INTERNATIONAL GONGRESS 82 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z">(2.3)|PH,ξ(φ)|2=|Sπ|−1ΔL(12,πE)L(1,π,Ad)∏vPH,ξ,v♮(φv,φv)
where Δ = i = 1 n L ( i , η E / F i ) Δ = ∏ i = 1 n   L i , η E / F i Delta=prod_(i=1)^(n)L(i,eta_(E//F)^(i))\Delta=\prod_{i=1}^{n} L\left(i, \eta_{E / F}^{i}\right)Δ=∏i=1nL(i,ηE/Fi) and L ( s , π , A d ) = v L ( s , π v , A d ) L ( s , Ï€ , A d ) = ∏ v   L s , Ï€ v , A d L(s,pi,Ad)=prod_(v)L(s,pi_(v),Ad)L(s, \pi, A d)=\prod_{v} L\left(s, \pi_{v}, A d\right)L(s,Ï€,Ad)=∏vL(s,Ï€v,Ad) denotes the completed adjoint L L LLL-function of π Ï€ pi\piÏ€.
Note that at a formal level, that is, formally expanding L L LLL-functions as Euler products outside the range of convergence, the above formula can be rewritten in the more compact way as
(2.4) | P H , ξ ( φ ) | 2 = | S π | 1 v P H , ξ , v ( φ v , φ v ) (2.4) P H , ξ ( φ ) 2 = S Ï€ − 1 ∏ v ′   P H , ξ , v φ v , φ v {:(2.4)|P_(H,xi)(varphi)|^(2)=|S_(pi)|^(-1)prod_(v)^(')P_(H,xi,v)(varphi_(v),varphi_(v)):}\begin{equation*} \left|\mathcal{P}_{H, \xi}(\varphi)\right|^{2}=\left|S_{\pi}\right|^{-1} \prod_{v}^{\prime} \mathscr{P}_{H, \xi, v}\left(\varphi_{v}, \varphi_{v}\right) \tag{2.4} \end{equation*}(2.4)|PH,ξ(φ)|2=|SÏ€|−1∏v′PH,ξ,v(φv,φv)
where the prime symbol on the product sign indicates that it is not convergent and has to be suitably reinterpreted "in the sense of L L LLL-functions" as identity (2.3).
Thanks to the work of many authors that we are going to summarize in the next sections, it is now relatively easy to state the current status on these two conjectures:
Theorem 2.1. Both Conjectures 2.1 and 2.2 hold in full generality.
The rest of this paper is devoted to reviewing the long series of works leading to the above theorem. They all stem from a strategy originally proposed by Jacquet and Rallis [32] of comparing two relative trace formulae. Let us mention here that there has actually been other fruitful approaches to the global Gan-Gross-Prasad conjecture among which one of the most notable has been the method pioneered by Ginzburg-Jiang-Rallis [25] using automorphic descent and that has recently seen much development with the work [33] of Jiang and L. Zhang proving in full generality the implication (2) ⇒ =>\Rightarrow⇒ (1) of Conjecture 2.1.

2.2. The approach of Jacquet-Rallis

In [32], Jacquet and Rallis have proposed a way to attack the Gan-Gross-Prasad conjecture for unitary groups through a comparison of relative trace formulae. They only consider the case where dim ( W ) = dim ( V ) 1 dim ⁡ ( W ) = dim ⁡ ( V ) − 1 dim(W)=dim(V)-1\operatorname{dim}(W)=\operatorname{dim}(V)-1dim⁡(W)=dim⁡(V)−1 (in which case H = U ( W ) H = U ( W ) H=U(W)H=U(W)H=U(W) and the character ξ ξ xi\xiξ is trivial) and we assume throughout that this condition is satisfied. The global relative trace formulae considered here are powerful analytic tools introduced originally by Jacquet and that relate automorphic periods to more geometric distributions known as relative orbital integrals.
Let us be more specific in the case at hand. For f C c ( G ( A F ) ) f ∈ C c ∞ G A F f inC_(c)^(oo)(G(A_(F)))f \in C_{c}^{\infty}\left(G\left(\mathbb{A}_{F}\right)\right)f∈Cc∞(G(AF)), a global test function, we let
K f ( x , y ) = γ G ( F ) f ( x 1 γ y ) , x , y G ( F ) G ( A F ) K f ( x , y ) = ∑ γ ∈ G ( F )   f x − 1 γ y , x , y ∈ G ( F ) ∖ G A F K_(f)(x,y)=sum_(gamma in G(F))f(x^(-1)gamma y),quad x,y in G(F)\\G(A_(F))K_{f}(x, y)=\sum_{\gamma \in G(F)} f\left(x^{-1} \gamma y\right), \quad x, y \in G(F) \backslash G\left(\mathbb{A}_{F}\right)Kf(x,y)=∑γ∈G(F)f(x−1γy),x,y∈G(F)∖G(AF)
be its automorphic kernel which describes the operator R ( f ) R ( f ) R(f)R(f)R(f) of right convolution by f f fff on the space of automorphic forms. The first trace formula introduced by Jacquet and Rallis is formally obtained by expanding the (usually divergent) expression
(2.5) J ( f ) = [ H ] × [ H ] K f ( h 1 , h 2 ) d h 1 d h 2 (2.5) J ( f ) = ∫ [ H ] × [ H ]   K f h 1 , h 2 d h 1 d h 2 {:(2.5)J(f)=int_([H]xx[H])K_(f)(h_(1),h_(2))dh_(1)dh_(2):}\begin{equation*} J(f)=\int_{[H] \times[H]} K_{f}\left(h_{1}, h_{2}\right) d h_{1} d h_{2} \tag{2.5} \end{equation*}(2.5)J(f)=∫[H]×[H]Kf(h1,h2)dh1dh2
in two different ways. More precisely, but still at a formal level, this distribution can be expanded as
(2.6) + δ H ( F ) G r s ( F ) / H ( F ) O ( δ , f ) = J ( f ) = φ A cusp ( G ) P H ( R ( f ) φ ) P H ( φ ) ¯ + (2.6) ⋯ + ∑ δ ∈ H ( F ) ∖ G r s ( F ) / H ( F )   O ( δ , f ) = J ( f ) = ∑ φ ∈ A cusp  ( G )   P H ( R ( f ) φ ) P H ( φ ) ¯ + ⋯ {:(2.6)cdots+sum_(delta in H(F)\\G_(rs)(F)//H(F))O(delta","f)=J(f)=sum_(varphi inA_("cusp ")(G))P_(H)(R(f)varphi) bar(P_(H)(varphi))+cdots:}\begin{equation*} \cdots+\sum_{\delta \in H(F) \backslash G_{\mathrm{rs}}(F) / H(F)} O(\delta, f)=J(f)=\sum_{\varphi \in \mathscr{A}_{\text {cusp }}(G)} \mathcal{P}_{H}(R(f) \varphi) \overline{\mathcal{P}_{H}(\varphi)}+\cdots \tag{2.6} \end{equation*}(2.6)⋯+∑δ∈H(F)∖Grs(F)/H(F)O(δ,f)=J(f)=∑φ∈Acusp (G)PH(R(f)φ)PH(φ)¯+⋯
where the right sum runs over an orthonormal basis for the space of cuspidal automorphic forms whereas the left sum is indexed by the so-called regular semisimple double cosets of H ( F ) H ( F ) H(F)H(F)H(F) in G ( F ) G ( F ) G(F)G(F)G(F). Here, an element δ G δ ∈ G delta in G\delta \in Gδ∈G is called (relatively) regular semisimple if its stabilizer under the H × H H × H H xx HH \times HH×H-action is trivial and the corresponding orbit is (Zariski) closed. We denote by G r s G r s G_(rs)G_{r s}Grs the nonempty Zariski open subset of regular semisimple elements and for δ G r s ( F ) δ ∈ G r s ( F ) delta inG_(rs)(F)\delta \in G_{\mathrm{rs}}(F)δ∈Grs(F),
O ( δ , f ) = H ( A F ) × H ( A F ) f ( h 1 δ h 2 ) d h 1 d h 2 O ( δ , f ) = ∫ H A F × H A F   f h 1 δ h 2 d h 1 d h 2 O(delta,f)=int_(H(A_(F))xx H(A_(F)))f(h_(1)deltah_(2))dh_(1)dh_(2)O(\delta, f)=\int_{H\left(\mathbb{A}_{F}\right) \times H\left(\mathbb{A}_{F}\right)} f\left(h_{1} \delta h_{2}\right) d h_{1} d h_{2}O(δ,f)=∫H(AF)×H(AF)f(h1δh2)dh1dh2
denotes the corresponding relative orbital integral of f f fff at δ δ delta\deltaδ. The left suspension points in (2.6) represent the remaining contributions from singular orbits whereas the right suspension points indicate the contribution of the continuous spectrum (both of which are somehow responsible for the divergence of the original expression (2.5)).
The second trace formula introduced by Jacquet and Rallis has to do with the following triple of groups:
H 1 = Res E / F G L n , E G = Res E / F G L n + 1 , E × Res E / F G L n , E H 2 = G L n + 1 , F × G L n , F , H 1 = Res E / F ⁡ G L n , E ↪ G ′ = Res E / F ⁡ G L n + 1 , E × Res E / F ⁡ G L n , E ↩ H 2 = G L n + 1 , F × G L n , F , {:[H_(1)=Res_(E//F)GL_(n,E)↪G^(')=Res_(E//F)GL_(n+1,E)xxRes_(E//F)GL_(n,E)↩H_(2)],[=GL_(n+1,F)xxGL_(n,F)","]:}\begin{aligned} H_{1} & =\operatorname{Res}_{E / F} \mathrm{GL}_{n, E} \hookrightarrow G^{\prime}=\operatorname{Res}_{E / F} \mathrm{GL}_{n+1, E} \times \operatorname{Res}_{E / F} \mathrm{GL}_{n, E} \hookleftarrow H_{2} \\ & =\mathrm{GL}_{n+1, F} \times \mathrm{GL}_{n, F}, \end{aligned}H1=ResE/F⁡GLn,E↪G′=ResE/F⁡GLn+1,E×ResE/F⁡GLn,E↩H2=GLn+1,F×GLn,F,
where n = dim ( W ) n = dim ⁡ ( W ) n=dim(W)n=\operatorname{dim}(W)n=dim⁡(W), the first embedding is the diagonal one and the second embedding is the natural one. Note that G G ′ G^(')G^{\prime}G′ is the group on which the base-change π E Ï€ E pi_(E)\pi_{E}Ï€E "lives." For f C c ( G ( A F ) ) f ′ ∈ C c ∞ G ′ A F f^(')inC_(c)^(oo)(G^(')(A_(F)))f^{\prime} \in C_{c}^{\infty}\left(G^{\prime}\left(\mathbb{A}_{F}\right)\right)f′∈Cc∞(G′(AF)), we write (again formally)
(2.7) I ( f ) = [ H 1 ] × [ H 2 ] K f ( h 1 , h 2 ) η ( h 2 ) d h 1 d h 2 (2.7) I f ′ = ∫ H 1 × H 2   K f ′ h 1 , h 2 η h 2 d h 1 d h 2 {:(2.7)I(f^('))=int_([H_(1)]xx[H_(2)])K_(f^('))(h_(1),h_(2))eta(h_(2))dh_(1)dh_(2):}\begin{equation*} I\left(f^{\prime}\right)=\int_{\left[H_{1}\right] \times\left[H_{2}\right]} K_{f^{\prime}}\left(h_{1}, h_{2}\right) \eta\left(h_{2}\right) d h_{1} d h_{2} \tag{2.7} \end{equation*}(2.7)I(f′)=∫[H1]×[H2]Kf′(h1,h2)η(h2)dh1dh2
where K f K f ′ K_(f^('))K_{f^{\prime}}Kf′ is the automorphic kernel of f f ′ f^(')f^{\prime}f′ and η : [ H 2 ] { ± 1 } η : H 2 → { ± 1 } eta:[H_(2)]rarr{+-1}\eta:\left[H_{2}\right] \rightarrow\{ \pm 1\}η:[H2]→{±1} is the automorphic character defined by η ( g n , g n + 1 ) = η E / F ( det g n ) n + 1 η E / F ( det g n + 1 ) n η g n , g n + 1 = η E / F det ⁡ g n n + 1 η E / F det ⁡ g n + 1 n eta(g_(n),g_(n+1))=eta_(E//F)(det g_(n))^(n+1)eta_(E//F)(det g_(n+1))^(n)\eta\left(g_{n}, g_{n+1}\right)=\eta_{E / F}\left(\operatorname{det} g_{n}\right)^{n+1} \eta_{E / F}\left(\operatorname{det} g_{n+1}\right)^{n}η(gn,gn+1)=ηE/F(det⁡gn)n+1ηE/F(det⁡gn+1)n. This formal distribution can be analogously expanded as
(2.8) + γ H 1 ( F ) G r s ( F ) / H 2 ( F ) O η ( γ , f ) = I ( f ) = φ A cusp ( G ) P H 1 ( R ( f ) φ ) P H 2 , η ( φ ) ¯ + (2.8) ⋯ + ∑ γ ∈ H 1 ( F ) ∖ G r s ′ ( F ) / H 2 ( F )   O η γ , f ′ = I f ′ = ∑ φ ∈ A cusp  G ′   P H 1 R f ′ φ P H 2 , η ( φ ) ¯ + ⋯ {:(2.8)cdots+sum_(gamma inH_(1)(F)\\G_(rs)^(')(F)//H_(2)(F))O_(eta)(gamma,f^('))=I(f^('))=sum_(varphi inA_("cusp ")(G^(')))P_(H_(1))(R(f^('))varphi) bar(P_(H_(2),eta)(varphi))+cdots:}\begin{equation*} \cdots+\sum_{\gamma \in H_{1}(F) \backslash G_{\mathrm{rs}}^{\prime}(F) / H_{2}(F)} O_{\eta}\left(\gamma, f^{\prime}\right)=I\left(f^{\prime}\right)=\sum_{\varphi \in \mathcal{A}_{\text {cusp }}\left(G^{\prime}\right)} \mathcal{P}_{H_{1}}\left(R\left(f^{\prime}\right) \varphi\right) \overline{\mathcal{P}_{H_{2}, \eta}(\varphi)}+\cdots \tag{2.8} \end{equation*}(2.8)⋯+∑γ∈H1(F)∖Grs′(F)/H2(F)Oη(γ,f′)=I(f′)=∑φ∈Acusp (G′)PH1(R(f′)φ)PH2,η(φ)¯+⋯
where G r s G r s ′ G_(rs)^(')G_{r s}^{\prime}Grs′ stands for the open subset of regular and semisimple elements under the H 1 × H 2 H 1 × H 2 H_(1)xxH_(2)H_{1} \times H_{2}H1×H2 action, the relative orbital integrals are given by
O η ( γ , f ) = H 1 ( A F ) × H 2 ( A F ) f ( h 1 γ h 2 ) η ( h 2 ) d h 1 d h 2 O η γ , f ′ = ∫ H 1 A F × H 2 A F   f ′ h 1 γ h 2 η h 2 d h 1 d h 2 O_(eta)(gamma,f^('))=int_(H_(1)(A_(F))xxH_(2)(A_(F)))f^(')(h_(1)gammah_(2))eta(h_(2))dh_(1)dh_(2)O_{\eta}\left(\gamma, f^{\prime}\right)=\int_{H_{1}\left(\mathbb{A}_{F}\right) \times H_{2}\left(\mathbb{A}_{F}\right)} f^{\prime}\left(h_{1} \gamma h_{2}\right) \eta\left(h_{2}\right) d h_{1} d h_{2}Oη(γ,f′)=∫H1(AF)×H2(AF)f′(h1γh2)η(h2)dh1dh2
and P H 1 , P H 2 , η P H 1 , P H 2 , η P_(H_(1)),P_(H_(2),eta)\mathscr{P}_{H_{1}}, \mathscr{P}_{H_{2}, \eta}PH1,PH2,η denote the automorphic period integrals over [ H 1 ] H 1 [H_(1)]\left[H_{1}\right][H1] and [ H 2 ] H 2 [H_(2)]\left[H_{2}\right][H2] twisted by η η eta\etaη, respectively.
The discussion so far is, of course, oversimplifying and ignoring serious analytical and convergence issues (we will come back to this later). However, as a motivation for considering this relative trace formula on G G ′ G^(')G^{\prime}G′, we have the following results on automorphic periods:
  • The period P H 1 P H 1 P_(H_(1))\mathscr{P}_{H_{1}}PH1 is a Rankin-Selberg period studied by Jacquet-PiatetskiiShapiro-Shalika that essentially represents the central value L ( 1 2 , Π ) L 1 2 , Π L((1)/(2),Pi)L\left(\frac{1}{2}, \Pi\right)L(12,Π) on Π A cusp ( G ) Π ↪ A cusp  G ′ Pi↪A_("cusp ")(G^('))\Pi \hookrightarrow \mathscr{A}_{\text {cusp }}\left(G^{\prime}\right)Π↪Acusp (G′);
  • The period P H 2 , η P H 2 , η P_(H_(2),eta)\mathcal{P}_{H_{2}, \eta}PH2,η was studied by Rallis and Flicker who have shown that it detects exactly the cuspidal automorphic Π Î  Pi\PiΠ 's that come by base-change from G G GGG (i.e., it is nonzero precisely on those cuspidal representations of the form π E Ï€ E pi_(E)\pi_{E}Ï€E, for π Ï€ pi\piÏ€ a cuspidal automorphic representation of G G GGG ).
Thus, on a very formal and sketchy sense, the Gan-Gross-Prasad conjecture implies that the spectral sides of I ( f ) I f ′ I(f^('))I\left(f^{\prime}\right)I(f′) should somehow "match" that of J ( f ) J ( f ) J(f)J(f)J(f). The idea of Jacquet and Rallis was to make precise the existence of such a comparison, from which the global GanGross-Prasad conjecture was eventually to be deduced, by equalling the geometric sides term by term. As a first step, they define a correspondence of orbits, which here takes the form of a natural embedding between regular semisimple cosets
(2.9) H ( k ) G r s ( k ) / H ( k ) H 1 ( k ) G r s ( k ) / H 2 ( k ) , δ γ (2.9) H ( k ) ∖ G r s ( k ) / H ( k ) ↪ H 1 ( k ) ∖ G r s ′ ( k ) / H 2 ( k ) , δ ↦ γ {:(2.9)H(k)\\G_(rs)(k)//H(k)↪H_(1)(k)\\G_(rs)^(')(k)//H_(2)(k)","quad delta|->gamma:}\begin{equation*} H(k) \backslash G_{\mathrm{rs}}(k) / H(k) \hookrightarrow H_{1}(k) \backslash G_{\mathrm{rs}}^{\prime}(k) / H_{2}(k), \quad \delta \mapsto \gamma \tag{2.9} \end{equation*}(2.9)H(k)∖Grs(k)/H(k)↪H1(k)∖Grs′(k)/H2(k),δ↦γ
for every field extension k / F k / F k//Fk / Fk/F. Using this correspondence, they then introduced a related notion of local transfer (or matching): for a place v v vvv of F F FFF, two test functions f v C c ( G v ) f v ∈ C c ∞ G v f_(v)inC_(c)^(oo)(G_(v))f_{v} \in C_{c}^{\infty}\left(G_{v}\right)fv∈Cc∞(Gv) and f v C c ( G v ) f v ′ ∈ C c ∞ G v ′ f_(v)^(')inC_(c)^(oo)(G_(v)^('))f_{v}^{\prime} \in C_{c}^{\infty}\left(G_{v}^{\prime}\right)fv′∈Cc∞(Gv′) are said to be transfers of each other (simply denoted by f v f v f v ↔ f v ′ f_(v)harrf_(v)^(')f_{v} \leftrightarrow f_{v}^{\prime}fv↔fv′ for short) if for every δ H ( F v ) G r s ( F v ) / H ( F v ) δ ∈ H F v ∖ G r s F v / H F v delta in H(F_(v))\\G_(rs)(F_(v))//H(F_(v))\delta \in H\left(F_{v}\right) \backslash G_{\mathrm{rs}}\left(F_{v}\right) / H\left(F_{v}\right)δ∈H(Fv)∖Grs(Fv)/H(Fv) we have an identity
(2.10) O ( δ , f v ) = Ω v ( γ ) O η v ( γ , f v ) (2.10) O δ , f v = Ω v ( γ ) O η v γ , f v ′ {:(2.10)O(delta,f_(v))=Omega_(v)(gamma)O_(eta_(v))(gamma,f_(v)^(')):}\begin{equation*} O\left(\delta, f_{v}\right)=\Omega_{v}(\gamma) O_{\eta_{v}}\left(\gamma, f_{v}^{\prime}\right) \tag{2.10} \end{equation*}(2.10)O(δ,fv)=Ωv(γ)Oηv(γ,fv′)
where γ H 1 ( F v ) G r s ( F v ) / H 2 ( F v ) γ ∈ H 1 F v ∖ G r s ′ F v / H 2 F v gamma inH_(1)(F_(v))\\G_(rs)^(')(F_(v))//H_(2)(F_(v))\gamma \in H_{1}\left(F_{v}\right) \backslash G_{r s}^{\prime}\left(F_{v}\right) / H_{2}\left(F_{v}\right)γ∈H1(Fv)∖Grs′(Fv)/H2(Fv) is the image of δ δ delta\deltaδ by the above correspondence, O ( δ , f v ) O δ , f v O(delta,f_(v))O\left(\delta, f_{v}\right)O(δ,fv) and O η v ( γ , f v ) O η v γ , f v ′ O_(eta_(v))(gamma,f_(v)^('))O_{\eta_{v}}\left(\gamma, f_{v}^{\prime}\right)Oηv(γ,fv′) are local relative orbital integrals defined in the same way as their global counterparts (replacing in the domain of integration, adelic groups by the corresponding local groups) and γ Ω v ( γ ) γ ↦ Ω v ( γ ) gamma|->Omega_(v)(gamma)\gamma \mapsto \Omega_{v}(\gamma)γ↦Ωv(γ) is a certain transfer factor which in particular has the effect of making the right-hand side above H 1 ( F v ) × H 2 ( F v ) H 1 F v × H 2 F v H_(1)(F_(v))xxH_(2)(F_(v))H_{1}\left(F_{v}\right) \times H_{2}\left(F_{v}\right)H1(Fv)×H2(Fv)-invariant in γ γ gamma\gammaγ.
As in the usual paradigm of endoscopy, to make this notion useful and allow for a global comparison we basically need two local ingredients: first the existence of local transfer (i.e., for every f v C c ( G v ) f v ∈ C c ∞ G v f_(v)inC_(c)^(oo)(G_(v))f_{v} \in C_{c}^{\infty}\left(G_{v}\right)fv∈Cc∞(Gv) there exists f v C c ( G v ) f v ′ ∈ C c ∞ G v ′ f_(v)^(')inC_(c)^(oo)(G_(v)^('))f_{v}^{\prime} \in C_{c}^{\infty}\left(G_{v}^{\prime}\right)fv′∈Cc∞(Gv′) such that f v f v f v ↔ f v ′ f_(v)harrf_(v)^(')f_{v} \leftrightarrow f_{v}^{\prime}fv↔fv′ and conversely, every f v f v ′ f_(v)^(')f_{v}^{\prime}fv′ admits a transfer f v f v f_(v)f_{v}fv ) and then a fundamental lemma (saying, at least, that 1 G ( O v ) 1 G ( O v ) 1 G O v ↔ 1 G ′ O v 1_(G(O_(v)))harr1_(G^(')(O_(v)))\mathbf{1}_{G\left(\mathcal{O}_{v}\right)} \leftrightarrow \mathbf{1}_{G^{\prime}\left(\mathcal{O}_{v}\right)}1G(Ov)↔1G′(Ov) for almost all v ) v {:v)\left.v\right)v).

3. COMPARISON: LOCAL TRANSFER AND FUNDAMENTAL LEMMA

A first breakthrough on the Jacquet-Rallis approach to the Gan-Gross-Prasad conjecture was made in [57] by Wei Zhang who proved the existence of the local transfer at all non-Archimedean places. His strategy for doing so roughly goes as follows:
  • The first step is to reduce to a statement on Lie algebras using some avatar of the exponential map (also known as Cayley map): we are then left with proving the existence of a similar transfer between the orbital integrals for the adjoint action of U ( W v ) U W v U(W_(v))U\left(W_{v}\right)U(Wv) on u ( V v ) = Lie ( U ( V v ) ) u V v = Lie ⁡ U V v u(V_(v))=Lie(U(V_(v)))\mathfrak{u}\left(V_{v}\right)=\operatorname{Lie}\left(U\left(V_{v}\right)\right)u(Vv)=Lie⁡(U(Vv)) and for the adjoint action of G L n ( F v ) G L n F v GL_(n)(F_(v))\mathrm{GL}_{n}\left(F_{v}\right)GLn(Fv) on g n + 1 ( F v ) g n + 1 F v g_(n+1)(F_(v))\mathfrak{g}_{n+1}\left(F_{v}\right)gn+1(Fv).
  • Then, a crucial ingredient in Zhang's proof is to show that the transfer at the Lie algebra level essentially commutes (i.e., up to some explicit multiplicative
    constants) with 3 different partial Fourier transforms F 1 , F 2 F 1 , F 2 F_(1),F_(2)\mathscr{F}_{1}, \mathscr{F}_{2}F1,F2, and F 3 F 3 F_(3)\mathscr{F}_{3}F3 that can naturally be defined on the two spaces C c ( u ( V v ) ) , C c ( g l n + 1 ( F v ) ) C c ∞ u V v , C c ∞ g l n + 1 F v C_(c)^(oo)(u(V_(v))),C_(c)^(oo)(gl_(n+1)(F_(v)))C_{c}^{\infty}\left(\mathfrak{u}\left(V_{v}\right)\right), C_{c}^{\infty}\left(\mathfrak{g l}_{n+1}\left(F_{v}\right)\right)Cc∞(u(Vv)),Cc∞(gln+1(Fv)). One of them, that we will denote by F 1 F 1 F_(1)\mathscr{F}_{1}F1, is the Fourier transform with respect to "the last row and column" on g l n + 1 ( F v ) g l n + 1 F v gl_(n+1)(F_(v))\mathfrak{g l}_{n+1}\left(F_{v}\right)gln+1(Fv) or u ( V v ) u V v u(V_(v))\mathfrak{u}\left(V_{v}\right)u(Vv) when realizing the latter in matrix form using a basis adapted to the decomposition V v = W v W v V v = W v ⊕ W v ⊥ V_(v)=W_(v)o+W_(v)^(_|_)V_{v}=W_{v} \oplus W_{v}^{\perp}Vv=Wv⊕Wv⊥. (Recall that we are assuming that dim ( W v ) = 1 dim ⁡ W v ⊥ = 1 dim(W_(v)^(_|_))=1\operatorname{dim}\left(W_{v}^{\perp}\right)=1dim⁡(Wv⊥)=1.) For this, Zhang develops some relative trace formulae for the aforementioned actions on g l n + 1 ( F v ) g l n + 1 F v gl_(n+1)(F_(v))\mathfrak{g l}_{n+1}\left(F_{v}\right)gln+1(Fv) and u ( V v ) u V v u(V_(v))\mathfrak{u}\left(V_{v}\right)u(Vv) and combines them with a clever induction argument.
  • Finally, the proof of the existence of transfer on Lie algebras is obtained by combining the second step with a certain uncertainty principle due to Aizenbud [1], which allows reducing the construction of the transfer to functions that are supported away from the relative nilpotent cones (i.e., the set of elements whose orbit closure contains an element of the center of the Lie algebra), as well as a standard descent argument whose essence goes back to Harish-Chandra.
It is noteworthy to mention that this result was subsequently extended, following the same strategy, by H. Xue [53] to Archimedean places, although the final result there is slightly weaker. (More precisely, Xue was only able to show that a dense subspace of test functions admit a transfer but also observed that it is sufficient for all expected applications.)
The Jacquet-Rallis fundamental lemma for its part, was proven earlier by Yun [55] in the case of fields of positive characteristic following and adapting the geometriccohomological approach based on Hitchin fibrations that was developed by N g ô N g ô Ngô\mathrm{Ngô}Ngô in the context of the endoscopic fundamental lemma. This result was then transferred to fields of characteristic zero, but of sufficiently large residual characteristic, using model-theoretic techniques by Julia Gordon in the appendix of [55].
Later, in [14], I found a completely new and elementary proof of this fundamental lemma. The argument, despite that of Gordon-Yun, works directly in characteristic zero and is purely based on techniques from harmonic analysis. Thus, we have:
Theorem 3.1 (Yun-Gordon, Beuzart-Plessis). Let v v vvv be a place of F F FFF of residue characteristic not 2 that is unramified in E E EEE and assume that the Hermitian spaces W v , W v W v , W v ⊥ W_(v),W_(v)^(_|_)W_{v}, W_{v}^{\perp}Wv,Wv⊥ both admit self-dual lattices L v W L v W L_(v)^(W)L_{v}^{W}LvW and L v W L v W ⊥ L_(v)^(W^(_|_))L_{v}^{W^{\perp}}LvW⊥. Set L v = L v W L v W L v = L v W ⊕ L v W ⊥ L_(v)=L_(v)^(W)o+L_(v)^(W^(_|_))L_{v}=L_{v}^{W} \oplus L_{v}^{W^{\perp}}Lv=LvW⊕LvW⊥ (a self-dual lattice in V v V v V_(v)V_{v}Vv ) and K v = Stab G v ( L v × L v W ) K v = Stab G v ⁡ L v × L v W K_(v)=Stab_(G_(v))(L_(v)xxL_(v)^(W))K_{v}=\operatorname{Stab}_{G_{v}}\left(L_{v} \times L_{v}^{W}\right)Kv=StabGv⁡(Lv×LvW) for the stabilizer in G v = U ( V v ) × U ( W v ) G v = U V v × U W v G_(v)=U(V_(v))xx U(W_(v))G_{v}=U\left(V_{v}\right) \times U\left(W_{v}\right)Gv=U(Vv)×U(Wv) of the lattices L v L v L_(v)L_{v}Lv and L v W L v W L_(v)^(W)L_{v}^{W}LvW (a hyperspecial compact subgroup of G v G v G_(v)G_{v}Gv ). Then, setting K v = GL n + 1 ( O E v ) × K v ′ = GL n + 1 ⁡ O E v × K_(v)^(')=GL_(n+1)(O_(E_(v)))xxK_{v}^{\prime}=\operatorname{GL}_{n+1}\left(\mathcal{O}_{E_{v}}\right) \timesKv′=GLn+1⁡(OEv)× GL n ( O E v ) GL n ⁡ O E v GL_(n)(O_(E_(v)))\operatorname{GL}_{n}\left(\mathcal{O}_{E_{v}}\right)GLn⁡(OEv), we have 1 K v 1 K v 1 K v ↔ 1 K v ′ 1_(K_(v))harr1_(K_(v)^('))\mathbf{1}_{K_{v}} \leftrightarrow \mathbf{1}_{K_{v}^{\prime}}1Kv↔1Kv′.
More precisely, in [14] I proved a Lie algebra analog of the Jacquet-Rallis fundamental lemma (of which the original statement can easily be reduced; at least in residual characteristic not 2) stating that the relative orbital integrals of 1 u ( L v ) 1 u L v 1_(u(L_(v)))\mathbf{1}_{\mathfrak{u}\left(L_{v}\right)}1u(Lv) match those of 1 g l n + 1 ( O F v ) 1 g l n + 1 O F v 1_(gl_(n+1))(O_(F_(v)))\mathbf{1}_{\mathfrak{g} \mathfrak{l}_{n+1}}\left(\mathcal{O}_{F_{v}}\right)1gln+1(OFv) in a suitable sense (where u ( L v ) u L v u(L_(v))u\left(L_{v}\right)u(Lv) denotes the lattice in u ( V v ) u V v u(V_(v))u\left(V_{v}\right)u(Vv) stabilizing L v L v L_(v)L_{v}Lv ). The argument is based on a hidden SL(2) symmetry involving a Weil representation. More specifically, we consider the Weil representations of SL ( 2 , F v ) SL ⁡ 2 , F v SL(2,F_(v))\operatorname{SL}\left(2, F_{v}\right)SL⁡(2,Fv) associated to the quadratic form q q qqq sending a
matrix of size n + 1 n + 1 n+1n+1n+1,
X = ( A b c λ ) X = A b c λ X=([A,b],[c,lambda])X=\left(\begin{array}{ll} A & b \\ c & \lambda \end{array}\right)X=(Abcλ)
either in g l n + 1 ( F v ) g l n + 1 F v gl_(n+1)(F_(v))\mathfrak{g l}_{n+1}\left(F_{v}\right)gln+1(Fv) or in u ( V v ) u V v u(V_(v))\mathfrak{u}\left(V_{v}\right)u(Vv), to q ( X ) = c b q ( X ) = c b q(X)=cbq(X)=c bq(X)=cb (where here, A A AAA is a square-matrix, b b bbb is a column vector, and c c ccc a row vector all of size n n nnn ). Using the aforementioned result of Zhang that the transfer commutes with the partial Fourier transform F 1 F 1 F_(1)\mathscr{F}_{1}F1, it can be shown that these representations descend to spaces of relative orbital integrals on C c ( u ( V v ) ) C c ∞ u V v C_(c)^(oo)(u(V_(v)))C_{c}^{\infty}\left(u\left(V_{v}\right)\right)Cc∞(u(Vv)) and C c ( g l n + 1 ( F v ) ) C c ∞ g l n + 1 F v C_(c)^(oo)(gl_(n+1)(F_(v)))C_{c}^{\infty}\left(\mathfrak{g l}_{n+1}\left(F_{v}\right)\right)Cc∞(gln+1(Fv)) and coincide on their intersections (identifying the spaces of regular semisimple orbits through a correspondence similar to (2.9)). Consider then the difference
Φ : X u ( V v ) r s / U ( W v ) O ( X , 1 u ( L v V ) ) ω v ( Y ) O η v ( Y , 1 g I n + 1 ( O F v ) ) Φ : X ∈ u V v r s / U W v ↦ O X , 1 u L v V − ω v ( Y ) O η v Y , 1 g I n + 1 O F v Phi:X inu(V_(v))_(rs)//U(W_(v))|->O(X,1_(u(L_(v)^(V))))-omega_(v)(Y)O_(eta_(v))(Y,1_(gI_(n+1)(O_(F_(v)))))\Phi: X \in \mathfrak{u}\left(V_{v}\right)_{\mathrm{rs}} / U\left(W_{v}\right) \mapsto O\left(X, \mathbf{1}_{\mathfrak{u}\left(L_{v}^{V}\right)}\right)-\omega_{v}(Y) O_{\eta_{v}}\left(Y, \mathbf{1}_{\mathfrak{g} \mathfrak{I}_{n+1}\left(\mathcal{O}_{F_{v}}\right)}\right)Φ:X∈u(Vv)rs/U(Wv)↦O(X,1u(LvV))−ωv(Y)Oηv(Y,1gIn+1(OFv))
where u ( V v ) rs u V v rs  u(V_(v))_("rs ")\mathfrak{u}\left(V_{v}\right)_{\text {rs }}u(Vv)rs  denotes the Lie algebra analog of the relative regular semisimple locus, Y Y YYY is the image of X X XXX by a correspondence of orbits u ( V v ) r s / U ( W v ) g l n + 1 ( F v ) r s / G L n ( F v ) u V v r s / U W v ↪ g l n + 1 F v r s / G L n F v u(V_(v))_(rs)//U(W_(v))↪gl_(n+1)(F_(v))_(rs)//GL_(n)(F_(v))\mathfrak{u}\left(V_{v}\right)_{\mathrm{rs}} / U\left(W_{v}\right) \hookrightarrow \mathfrak{g l}_{n+1}\left(F_{v}\right)_{\mathrm{rs}} / \mathrm{GL}_{n}\left(F_{v}\right)u(Vv)rs/U(Wv)↪gln+1(Fv)rs/GLn(Fv) similar to (2.9) and ω v ( Y ) ω v ( Y ) omega_(v)(Y)\omega_{v}(Y)ωv(Y) is the Lie algebra counterpart of the transfer factor. The fundamental lemma then states that Φ Î¦ Phi\PhiΦ is identically zero. The proof proceeds roughly in three steps:
  • First, we show that Φ ( X ) = 0 Φ ( X ) = 0 Phi(X)=0\Phi(X)=0Φ(X)=0 for | q ( X ) | 1 | q ( X ) | ⩾ 1 |q(X)| >= 1|q(X)| \geqslant 1|q(X)|⩾1. When | q ( X ) | = 1 | q ( X ) | = 1 |q(X)|=1|q(X)|=1|q(X)|=1, this requires an inductive argument on n n nnn. Moreover, this vanishing can be reformulated by saying that Φ Î¦ Phi\PhiΦ is fixed by the subgroup ( 1 p F v 1 0 1 ) 1 p F v − 1 0 1 ([1,p_(F_(v))^(-1)],[0,1])\left(\begin{array}{cc}1 & p_{F_{v}}^{-1} \\ 0 & 1\end{array}\right)(1pFv−101) through the Weil representation.
  • Secondly, we remark that Φ Î¦ Phi\PhiΦ is also fixed by w = ( 0 1 1 0 ) w = 0 − 1 1 0 w=([0,-1],[1,0])w=\left(\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right)w=(0−110). This comes from the fact that the action of w w www descends from the partial Fourier transform F 1 F 1 F_(1)\mathscr{F}_{1}F1 which leaves (for a suitable normalization) the functions 1 u ( L v V ) , 1 g n + 1 ( O F v ) 1 u L v V , 1 g n + 1 O F v 1_(u(L_(v)^(V))),1_(g_(n+1)(O_(F_(v))))\mathbf{1}_{\mathfrak{u}\left(L_{v}^{V}\right)}, \mathbf{1}_{\mathfrak{g}_{n+1}\left(\mathcal{O}_{F_{v}}\right)}1u(LvV),1gn+1(OFv) invariant.
  • Finally, as S L 2 ( F v ) S L 2 F v SL_(2)(F_(v))\mathrm{SL}_{2}\left(F_{v}\right)SL2(Fv) is generated by ( 1 p F v 1 0 1 ) 1 p F v − 1 0 1 ([1p_(F_(v))^(-1)],[0],[1])\left(\begin{array}{c}1 \mathcal{p}_{F_{v}}^{-1} \\ 0 \\ 1\end{array}\right)(1pFv−101) and w w www, we infer that Φ Î¦ Phi\PhiΦ is fixed by S L 2 ( F v ) S L 2 F v SL_(2)(F_(v))\mathrm{SL}_{2}\left(F_{v}\right)SL2(Fv) from which it is relatively straightforward to deduce Φ = 0 Φ = 0 Phi=0\Phi=0Φ=0.
It is also worth mentioning that in a very interesting work, Jingwei Xiao [51] has shown that the Jacquet-Rallis fundamental lemma implies the (usual) endocospic fundamental lemma for unitary groups. Thus, combining his argument with the proof outlined above yields a completely elementary proof of the Langlands-Shelstad fundamental lemma for unitary groups!
The two previous results on smooth transfer and the fundamental lemma are already enough to imply the Gan-Gross-Prasad Conjecture 2.1 under some local restrictions on the cuspidal representation π Ï€ pi\piÏ€ (originating from the use of simple versions of the Jacquet-Rallis trace formulae, allowing to bypass all convergence issues) as was done by W. Zhang in [57]. However, to derive the refinement of Conjecture 2.2 following the same strategy, we need an extra local ingredient relating the local periods of Ichino-Ikeda to similar local distributions associated to the Rankin-Selberg and Flicker-Rallis periods. More precisely, by the work of Jacquet-Piatetskii-Shapiro-Shalika, on the one hand, and Flicker-Rallis, on the other hand,
it is known that the two automorphic periods P H 1 P H 1 P_(H_(1))\mathscr{P}_{H_{1}}PH1 and P H 2 , η P H 2 , η P_(H_(2),eta)\mathscr{P}_{H_{2}, \eta}PH2,η admit factorizations of the form
(3.1) P H 1 ( φ ) = v P H 1 , v ( W φ , v ) (3.2) P H 2 , η ( φ ) = 1 4 v P H 2 , η , v ( W φ , v ) (3.1) P H 1 ( φ ) = ∏ v ′   P H 1 , v W φ , v (3.2) P H 2 , η ( φ ) = 1 4 ∏ v ′   P H 2 , η , v W φ , v {:[(3.1)P_(H_(1))(varphi)=prod_(v)^(')P_(H_(1),v)(W_(varphi,v))],[(3.2)P_(H_(2),eta)(varphi)=(1)/(4)prod_(v)^(')P_(H_(2),eta,v)(W_(varphi,v))]:}\begin{align*} \mathcal{P}_{H_{1}}(\varphi) & =\prod_{v}^{\prime} \mathcal{P}_{H_{1}, v}\left(W_{\varphi, v}\right) \tag{3.1}\\ \mathcal{P}_{H_{2}, \eta}(\varphi) & =\frac{1}{4} \prod_{v}^{\prime} \mathcal{P}_{H_{2}, \eta, v}\left(W_{\varphi, v}\right) \tag{3.2} \end{align*}(3.1)PH1(φ)=∏v′PH1,v(Wφ,v)(3.2)PH2,η(φ)=14∏v′PH2,η,v(Wφ,v)
for φ φ varphi\varphiφ a factorizable vector in a given cuspidal automorphic representation Π = Π n + 1 Π n Π = Π n + 1 ⊗ Π n Pi=Pi_(n+1)oxPi_(n)\Pi=\Pi_{n+1} \otimes \Pi_{n}Π=Πn+1⊗Πn of G ( A F ) G ′ A F G^(')(A_(F))G^{\prime}\left(\mathbb{A}_{F}\right)G′(AF), where
W φ ( g ) = [ N ] φ ( u g ) ψ ( u ) 1 d u = v W φ , v ( g v ) W φ ( g ) = ∫ N ′   φ ( u g ) ψ ′ ( u ) − 1 d u = ∏ v   W φ , v g v W_(varphi)(g)=int_([N^(')])varphi(ug)psi^(')(u)^(-1)du=prod_(v)W_(varphi,v)(g_(v))W_{\varphi}(g)=\int_{\left[N^{\prime}\right]} \varphi(u g) \psi^{\prime}(u)^{-1} d u=\prod_{v} W_{\varphi, v}\left(g_{v}\right)Wφ(g)=∫[N′]φ(ug)ψ′(u)−1du=∏vWφ,v(gv)
denotes a factorization of the Whittaker function of φ φ varphi\varphiφ (here N N ′ N^(')N^{\prime}N′ stands for the standard maximal unipotent subgroup of G G ′ G^(')G^{\prime}G′ and ψ ψ ′ psi^(')\psi^{\prime}ψ′ is a nondegenerate character of [ N ] ) , P H 1 , v , P H 2 , η , v N ′ , P H 1 , v , P H 2 , η , v {:[N^(')]),P_(H_(1),v),P_(H_(2),eta,v)\left.\left[N^{\prime}\right]\right), \mathscr{P}_{H_{1}, v}, \mathscr{P}_{H_{2}, \eta, v}[N′]),PH1,v,PH2,η,v are explicit linear forms on the local Whittaker model W ( Π v , ψ v ) W Π v , ψ v ′ W(Pi_(v),psi_(v)^('))\mathcal{W}\left(\Pi_{v}, \psi_{v}^{\prime}\right)W(Πv,ψv′) of Π v Π v Pi_(v)\Pi_{v}Πv and the products in (3.1), (3.2) are to be regularized and understood "in the sense of L L LLL-functions" in a way similar to (2.4).
Based on the factorizations (3.1) and (3.2), the contribution of Π Î  Pi\PiΠ to the spectral expansion (2.8) can be shown to itself admit a factorization roughly as the product of local distributions (called relative characters) I Π v I Π v I_(Pi_(v))I_{\Pi_{v}}IΠv defined by
I Π v ( f v ) = W v W ( Π v , ψ v ) P H 1 , v ( Π v ( f v ) W v ) P H 2 , η , v ( W v ) ¯ , f v C c ( G v ) I Π v f v ′ = ∑ W v ∈ W Π v , ψ v ′   P H 1 , v Π v f v ′ W v P H 2 , η , v W v ¯ , f v ′ ∈ C c ∞ G v ′ I_(Pi_(v))(f_(v)^('))=sum_(W_(v)inW(Pi_(v),psi_(v)^(')))P_(H_(1),v)(Pi_(v)(f_(v)^('))W_(v)) bar(P_(H_(2),eta,v)(W_(v))),quadf_(v)^(')inC_(c)^(oo)(G_(v)^('))I_{\Pi_{v}}\left(f_{v}^{\prime}\right)=\sum_{W_{v} \in \mathcal{W}\left(\Pi_{v}, \psi_{v}^{\prime}\right)} \mathcal{P}_{H_{1}, v}\left(\Pi_{v}\left(f_{v}^{\prime}\right) W_{v}\right) \overline{\mathcal{P}_{H_{2}, \eta, v}\left(W_{v}\right)}, \quad f_{v}^{\prime} \in C_{c}^{\infty}\left(G_{v}^{\prime}\right)IΠv(fv′)=∑Wv∈W(Πv,ψv′)PH1,v(Πv(fv′)Wv)PH2,η,v(Wv)¯,fv′∈Cc∞(Gv′)
where the sum runs over a suitable orthonormal basis of the Whittaker model. On the other hand, from the Ichino-Ikeda Conjecture 2.2, we expect the contribution of π A cusp ( G ) Ï€ ↪ A cusp  ( G ) pi↪A_("cusp ")(G)\pi \hookrightarrow \mathcal{A}_{\text {cusp }}(G)π↪Acusp (G) to the spectral expansion of (2.6) to essentially factorize into the product of the local relative characters (where again the sum is taken over an orthonormal basis)
J π v ( f v ) = φ v π v P H , v ( π v ( f v ) φ v , φ v ) , f v C c ( G v ) J Ï€ v f v = ∑ φ v ∈ Ï€ v   P H , v Ï€ v f v φ v , φ v , f v ∈ C c ∞ G v J_(pi_(v))(f_(v))=sum_(varphi_(v)inpi_(v))P_(H,v)(pi_(v)(f_(v))varphi_(v),varphi_(v)),quadf_(v)inC_(c)^(oo)(G_(v))J_{\pi_{v}}\left(f_{v}\right)=\sum_{\varphi_{v} \in \pi_{v}} \mathcal{P}_{H, v}\left(\pi_{v}\left(f_{v}\right) \varphi_{v}, \varphi_{v}\right), \quad f_{v} \in C_{c}^{\infty}\left(G_{v}\right)JÏ€v(fv)=∑φv∈πvPH,v(Ï€v(fv)φv,φv),fv∈Cc∞(Gv)
In [56], W. Zhang has conjectured that the local Jacquet-Rallis transfer f v f v f v ↔ f v ′ f_(v)harrf_(v)^(')f_{v} \leftrightarrow f_{v}^{\prime}fv↔fv′ also satisfies certain precise spectral relations involving the above relative characters. This is exactly the extra local ingredient needed to finish the proof of the Ichino-Ikeda conjecture based on a comparison of the Jacquet-Rallis trace formulae. This conjecture was shown in [56] to hold for unramified and supercuspidal representations, and the method was further extended and amplified in [13], allowing to prove the conjecture for all (tempered) representations at non-Archimedean places. Later, in [15] I gave a better proof of this conjecture which also has the advantage of working uniformly at all places (including Archimedean ones). To state the result, we introduce some terminology/notation: for a place v v vvv of F F FFF and a smooth irreducible representation π v Ï€ v pi_(v)\pi_{v}Ï€v of G v G v G_(v)G_{v}Gv, we denote by π E , v Ï€ E , v pi_(E,v)\pi_{E, v}Ï€E,v the local base-change of π v Ï€ v pi_(v)\pi_{v}Ï€v, that is, the smooth irreducible representation of G v G v ′ G_(v)^(')G_{v}^{\prime}Gv′ whose L L LLL-parameter is given by composing that of π v Ï€ v pi_(v)\pi_{v}Ï€v with the natural embedding of L L LLL-groups L G v L G v L G v → L G v ′ ^(L)G_(v)rarr^(L)G_(v)^('){ }^{L} G_{v} \rightarrow{ }^{L} G_{v}^{\prime}LGv→LGv′, and, moreover, we say that π v Ï€ v pi_(v)\pi_{v}Ï€v is H v H v H_(v)H_{v}Hv-distinguished if Hom H v ( π v , C ) 0 Hom H v ⁡ Ï€ v , C ≠ 0 Hom_(H_(v))(pi_(v),C)!=0\operatorname{Hom}_{H_{v}}\left(\pi_{v}, \mathbb{C}\right) \neq 0HomHv⁡(Ï€v,C)≠0, that is, with the notation of Section 1.1, if the multiplicity m ( π v ) m Ï€ v m(pi_(v))m\left(\pi_{v}\right)m(Ï€v) equals 1 .
Theorem 3.2. There exist explicit local constants ( κ v ) v κ v v (kappa_(v))_(v)\left(\kappa_{v}\right)_{v}(κv)v indexed by the set of all places of F F FFF and satisfying the product formula v κ v = 1 ∏ v   κ v = 1 prod_(v)kappa_(v)=1\prod_{v} \kappa_{v}=1∏vκv=1 such that the following property is verified: for every place v v vvv, every tempered representation π v Ï€ v pi_(v)\pi_{v}Ï€v of G v G v G_(v)G_{v}Gv which is H v H v H_(v)H_{v}Hv-distinguished and every pair ( f v , f v ) C c ( G v ) × C c ( G v ) f v , f v ′ ∈ C c ∞ G v × C c ∞ G v ′ (f_(v),f_(v)^('))inC_(c)^(oo)(G_(v))xxC_(c)^(oo)(G_(v)^('))\left(f_{v}, f_{v}^{\prime}\right) \in C_{c}^{\infty}\left(G_{v}\right) \times C_{c}^{\infty}\left(G_{v}^{\prime}\right)(fv,fv′)∈Cc∞(Gv)×Cc∞(Gv′) of matching functions (that is, f v f v f v ↔ f v ′ f_(v)harrf_(v)^(')f_{v} \leftrightarrow f_{v}^{\prime}fv↔fv′ ), we have
(3.3) I π v , E ( f v ) = κ v J π v ( f v ) (3.3) I Ï€ v , E f v ′ = κ v J Ï€ v f v {:(3.3)I_(pi_(v,E))(f_(v)^('))=kappa_(v)J_(pi_(v))(f_(v)):}\begin{equation*} I_{\pi_{v, E}}\left(f_{v}^{\prime}\right)=\kappa_{v} J_{\pi_{v}}\left(f_{v}\right) \tag{3.3} \end{equation*}(3.3)IÏ€v,E(fv′)=κvJÏ€v(fv)
Moreover, the above identities characterize the Jacquet-Rallis transfer, that is, if two functions f v C c ( G v ) , f v C c ( G v ) f v ∈ C c ∞ G v , f v ′ ∈ C c ∞ G v ′ f_(v)inC_(c)^(oo)(G_(v)),f_(v)^(')inC_(c)^(oo)(G_(v)^('))f_{v} \in C_{c}^{\infty}\left(G_{v}\right), f_{v}^{\prime} \in C_{c}^{\infty}\left(G_{v}^{\prime}\right)fv∈Cc∞(Gv),fv′∈Cc∞(Gv′) satisfy (3.3) for every tempered irreducible representation π v Ï€ v pi_(v)\pi_{v}Ï€v of G v G v G_(v)G_{v}Gv that is H v H v H_(v)H_{v}Hv-distinguished, then these functions are transfers of each other.
The proof given in [15] of the above theorem is mainly based on another ingredient of independent interest which is an explicit Plancherel decomposition for the space G v / H 2 , v G v ′ / H 2 , v G_(v)^(')//H_(2,v)G_{v}^{\prime} / H_{2, v}Gv′/H2,v or rather, decomposing this quotient as a product in a natural way, for the symmetric variety G L n ( E v ) / G L n ( F v ) G L n E v / G L n F v GL_(n)(E_(v))//GL_(n)(F_(v))\mathrm{GL}_{n}\left(E_{v}\right) / \mathrm{GL}_{n}\left(F_{v}\right)GLn(Ev)/GLn(Fv). This spectral decomposition is roughly obtained by applying the Plancherel formula for the group G L n ( E v ) G L n E v GL_(n)(E_(v))\mathrm{GL}_{n}\left(E_{v}\right)GLn(Ev) to a family of zeta integrals, depending on a complex parameter s s sss, introduced by Flicker and Rallis [22] and that represents local Asai L L LLL-factors and taking the residue at s = 1 s = 1 s=1s=1s=1 of the resulting expression. We will not describe the exact process here, but just mention that this settles in the case at hand a general conjecture of Sakellaridis-Venkatesh [41] on the spectral decomposition of spherical varieties. This Plancherel formula is then used to write the explicit spectral expansion for a local analog of the Jacquet-Rallis trace formula (2.8) which is then compared with a local counterpart of the trace formula (2.6) yielding as a consequence Theorem 3.2 above. Moreover, as another byproduct of this local comparison, we also get a formula conjectured by Hiraga-Ichino-Ikeda for the formal degree of discrete series [29] in the case of unitary groups.

4. GLOBAL ANALYSIS OF JACQUET-RALLIS TRACE FORMULAE

With all the local ingredients explained in the previous section in place, the only remaining tasks to finish the program initiated by Jacquet and Rallis to prove the GanGross-Prasad and Ichino-Ikeda conjectures are global. More specifically, although simple versions of the Jacquet-Rallis trace formulae have been successfully used to establish these conjectures under some local restrictions [13,57], in order to detect all the relevant cuspidal representations of unitary groups, we need more refined versions of the geometric and spectral expansions of (2.6) and (2.8).
As a first important step in that direction, Zydor [58,59] has completely regularized the singular contributions to the geometric sides. We can summarize his main results as follows: for all test functions f C c ( G ( A F ) ) f ∈ C c ∞ G A F f inC_(c)^(oo)(G(A_(F)))f \in C_{c}^{\infty}\left(G\left(\mathbb{A}_{F}\right)\right)f∈Cc∞(G(AF)) and f C c ( G ( A F ) ) f ′ ∈ C c ∞ G ′ A F f^(')inC_(c)^(oo)(G^(')(A_(F)))f^{\prime} \in C_{c}^{\infty}\left(G^{\prime}\left(\mathbb{A}_{F}\right)\right)f′∈Cc∞(G′(AF)), there exist "canonical" regularization of the (usually divergent) integrals (2.5) and (2.7), that we will still denote by J ( f ) J ( f ) J(f)J(f)J(f) and I ( f ) I f ′ I(f^('))I\left(f^{\prime}\right)I(f′), as well as decompositions
(4.1) J ( f ) = δ ( H G / / H ) ( F ) O ( δ , f ) and I ( f ) = γ ( H 1 G / / H 2 ) ( F ) O η ( γ , f ) (4.1) J ( f ) = ∑ δ ∈ ( H ∖ G / / H ) ( F )   O ( δ , f )  and  I f ′ = ∑ γ ∈ H 1 ∖ G ′ / / H 2 ( F )   O η γ , f ′ {:(4.1)J(f)=sum_(delta in(H\\G////H)(F))O(delta","f)quad" and "quad I(f^('))=sum_(gamma in(H_(1)\\G^(')////H_(2))(F))O_(eta)(gamma,f^(')):}\begin{equation*} J(f)=\sum_{\delta \in(H \backslash G / / H)(F)} O(\delta, f) \quad \text { and } \quad I\left(f^{\prime}\right)=\sum_{\gamma \in\left(H_{1} \backslash G^{\prime} / / H_{2}\right)(F)} O_{\eta}\left(\gamma, f^{\prime}\right) \tag{4.1} \end{equation*}(4.1)J(f)=∑δ∈(H∖G//H)(F)O(δ,f) and I(f′)=∑γ∈(H1∖G′//H2)(F)Oη(γ,f′)
where H G / / H H ∖ G / / H H\\G////HH \backslash G / / HH∖G//H and H 1 G / / H 2 H 1 ∖ G ′ / / H 2 H_(1)\\G^(')////H_(2)H_{1} \backslash G^{\prime} / / H_{2}H1∖G′//H2 stand for the corresponding categorical quotients and O ( δ , ) , O η ( γ , ) O ( δ , â‹… ) , O η ( γ , â‹… ) O(delta,*),O_(eta)(gamma,*)O(\delta, \cdot), O_{\eta}(\gamma, \cdot)O(δ,â‹…),Oη(γ,â‹…) are distributions supported on the union of the adelic double cosets with images δ δ delta\deltaδ and γ γ gamma\gammaγ in ( H G / / H ) ( A F ) ( H ∖ G / / H ) A F (H\\G////H)(A_(F))(H \backslash G / / H)\left(\mathbb{A}_{F}\right)(H∖G//H)(AF) and ( H 1 G / / H 2 ) ( A F ) H 1 ∖ G ′ / / H 2 A F (H_(1)\\G^(')////H_(2))(A_(F))\left(H_{1} \backslash G^{\prime} / / H_{2}\right)\left(\mathbb{A}_{F}\right)(H1∖G′//H2)(AF), respectively, which coincide with the previously defined relative orbital integrals when δ δ delta\deltaδ and γ γ gamma\gammaγ are regular semisimple.
Zydor obtains these regularized orbital integrals by adapting a truncation procedure developed by Arthur in the context of the usual trace formula to the relative setting at hand. It should be emphasized that contrary to what happens with Arthur's trace formula, the resulting distributions are directly invariant (in a relative sense, that is, here under the natural action of H × H H × H H xx HH \times HH×H or H 1 × H 2 H 1 × H 2 H_(1)xxH_(2)H_{1} \times H_{2}H1×H2 ) and do not depend on any auxiliary choice (such as that of a maximal compact subgroup). It is in this sense that the regularizations of Zydor are really "canonical." It should be mentioned that another, different, approach to such regularization was proposed by Sakellaridis [40] in the context of general relative trace formulae. It is based on analyzing the exponents at infinity of generalized theta series together with a natural procedure to regularize integrals of multiplicative functions when the corresponding character is nontrivial.
Before we even consider the analogous, more subtle, regularization problem on the spectral side, there appears the natural question of how to compare the singular contributions to the refined geometric expansions of (4.1). This issue was completely resolved in a very long paper [20] by Chaudouard and Zydor. To state their main result, it is convenient to again consider the relevant pure inner forms of G G GGG (as defined in Section 1.1): for every Hermitian space W W ′ W^(')W^{\prime}W′ of the same dimension as W W WWW, we have a relevant pure inner form G W = G W ′ = G^(W^('))=G^{W^{\prime}}=GW′= U ( V ) × U ( W ) U V ′ × U W ′ U(V^('))xx U(W^('))U\left(V^{\prime}\right) \times U\left(W^{\prime}\right)U(V′)×U(W′) with its diagonal subgroup H W = U ( W ) H W ′ = U W ′ H^(W^('))=U(W^('))H^{W^{\prime}}=U\left(W^{\prime}\right)HW′=U(W′) where V = W W V ′ = W ′ ⊕ ⊥ W ⊥ V^(')=W^(')o+^(_|_)W^(_|_)V^{\prime}=W^{\prime} \oplus^{\perp} W^{\perp}V′=W′⊕⊥W⊥. Moreover, the correspondence of orbits (2.9) extends to an isomorphism between categorical quotients,
(4.2) H G / / H H 1 G / / H 2 (4.2) H ∖ G / / H ≃ H 1 ∖ G ′ / / H 2 {:(4.2)H\\G////H≃H_(1)\\G^(')////H_(2):}\begin{equation*} H \backslash G / / H \simeq H_{1} \backslash G^{\prime} / / H_{2} \tag{4.2} \end{equation*}(4.2)H∖G//H≃H1∖G′//H2
and for every W W ′ W^(')W^{\prime}W′ as before, H W G W / / H W H W ′ ∖ G W ′ / / H W ′ H^(W^('))\\G^(W^('))////H^(W^('))H^{W^{\prime}} \backslash G^{W^{\prime}} / / H^{W^{\prime}}HW′∖GW′//HW′ can naturally be identified with H G / / H H ∖ G / / H H\\G////HH \backslash G / / HH∖G//H. With these preliminaries, the main result of Chaudouard and Zydor can now be stated as follows:
Theorem 4.1 (Chaudouard-Zydor). Assume that f W = v f v W C c ( G W ( A F ) ) f W ′ = ∏ v   f v W ′ ∈ C c ∞ G W ′ A F f^(W^('))=prod_(v)f_(v)^(W^('))inC_(c)^(oo)(G^(W^('))(A_(F)))f^{W^{\prime}}=\prod_{v} f_{v}^{W^{\prime}} \in C_{c}^{\infty}\left(G^{W^{\prime}}\left(\mathbb{A}_{F}\right)\right)fW′=∏vfvW′∈Cc∞(GW′(AF)), where W W ′ W^(')W^{\prime}W′ runs over all isomorphism classes of Hermitian spaces of dimension n n nnn, and f = v f v C c ( G ( A F ) ) f ′ = ∏ v   f v ′ ∈ C c ∞ G ′ A F f^(')=prod_(v)f_(v)^(')inC_(c)^(oo)(G^(')(A_(F)))f^{\prime}=\prod_{v} f_{v}^{\prime} \in C_{c}^{\infty}\left(G^{\prime}\left(\mathbb{A}_{F}\right)\right)f′=∏vfv′∈Cc∞(G′(AF)) are factorizable test functions such that for every place v v vvv, and each W , f v W W ′ , f v W ′ W^('),f_(v)^(W^('))W^{\prime}, f_{v}^{W^{\prime}}W′,fvW′ and f v f v ′ f_(v)^(')f_{v}^{\prime}fv′ are Jacquet-Rallis transfers of each other (that is, f v W f v f v W ′ ↔ f v ′ f_(v)^(W^('))harrf_(v)^(')f_{v}^{W^{\prime}} \leftrightarrow f_{v}^{\prime}fvW′↔fv′ ). Then, for every δ ( H G / / H ) ( F ) δ ∈ ( H ∖ G / / H ) ( F ) delta in(H\\G////H)(F)\delta \in(H \backslash G / / H)(F)δ∈(H∖G//H)(F) with image γ ( H 1 G / / H 2 ) ( F ) γ ∈ H 1 ∖ G ′ / / H 2 ( F ) gamma in(H_(1)\\G^(')////H_(2))(F)\gamma \in\left(H_{1} \backslash G^{\prime} / / H_{2}\right)(F)γ∈(H1∖G′//H2)(F) by (4.2), we have
(4.3) W O ( δ , f W ) = O η ( γ , f ) (4.3) ∑ W ′   O δ , f W ′ = O η γ , f ′ {:(4.3)sum_(W^('))O(delta,f^(W^(')))=O_(eta)(gamma,f^(')):}\begin{equation*} \sum_{W^{\prime}} O\left(\delta, f^{W^{\prime}}\right)=O_{\eta}\left(\gamma, f^{\prime}\right) \tag{4.3} \end{equation*}(4.3)∑W′O(δ,fW′)=Oη(γ,f′)
It should be noted that when δ δ delta\deltaδ, hence also γ γ gamma\gammaγ, is regular semisimple, the left-hand sum in (4.3) only contains one nonidentically vanishing term but that in general more than one relevant pure inner forms can contribute. Also, the above result extends to nonfactorizable test functions, provided the wording is changed suitably.
The next natural step would be to develop regularized spectral expansions similar to (4.1). As a first result in that direction, Zydor has shown decompositions of the form
(4.4) J ( f ) = χ X ( G ) J χ ( f ) and I ( f ) = χ X ( G ) I χ ( f ) (4.4) J ( f ) = ∑ χ ∈ X ( G )   J χ ( f )  and  I f ′ = ∑ χ ′ ∈ X G ′   I χ ′ f ′ {:(4.4)J(f)=sum_(chi inX(G))J_(chi)(f)quad" and "quad I(f^('))=sum_(chi^(')in X(G^(')))I_(chi^('))(f^(')):}\begin{equation*} J(f)=\sum_{\chi \in \mathcal{X}(G)} J_{\chi}(f) \quad \text { and } \quad I\left(f^{\prime}\right)=\sum_{\chi^{\prime} \in X\left(G^{\prime}\right)} I_{\chi^{\prime}}\left(f^{\prime}\right) \tag{4.4} \end{equation*}(4.4)J(f)=∑χ∈X(G)Jχ(f) and I(f′)=∑χ′∈X(G′)Iχ′(f′)
where X ( G ) X ( G ) X(G)\mathcal{X}(G)X(G) and X ( G ) X G ′ X(G^('))\mathcal{X}\left(G^{\prime}\right)X(G′) stand for the set of cuspidal data of the groups G G GGG and G G ′ G^(')G^{\prime}G′ respectively, that is the sets of pairs ( M , σ ) ( M , σ ) (M,sigma)(M, \sigma)(M,σ) where M M MMM is a Levi subgroup (of G G GGG or G G ′ G^(')G^{\prime}G′ ) and σ σ sigma\sigmaσ is a cuspidal automorphic representation of M ( A F ) M A F M(A_(F))M\left(\mathbb{A}_{F}\right)M(AF) taken up to conjugacy (by G ( F ) G ( F ) G(F)G(F)G(F) or G ( F ) G ′ ( F ) G^(')(F)G^{\prime}(F)G′(F) ). According to Langlands theory of pseudo-Eisenstein series, these sets index natural equivariant Hilbertian decompositions:
L 2 ( [ G ] ) = χ X ( G ) L χ 2 ( [ G ] ) , L 2 ( [ G ] ) = χ X ( G ) L χ 2 ( [ G ] ) L 2 ( [ G ] ) = ⨁ χ ∈ X ( G )   L χ 2 ( [ G ] ) , L 2 G ′ = ⨁ χ ′ ∈ X G ′   L χ ′ 2 G ′ L^(2)([G])=bigoplus_(chi inX(G))L_(chi)^(2)([G]),quadL^(2)([G^(')])=bigoplus_(chi^(')inX(G^(')))L_(chi^('))^(2)([G^(')])L^{2}([G])=\bigoplus_{\chi \in \mathcal{X}(G)} L_{\chi}^{2}([G]), \quad L^{2}\left(\left[G^{\prime}\right]\right)=\bigoplus_{\chi^{\prime} \in \mathcal{X}\left(G^{\prime}\right)} L_{\chi^{\prime}}^{2}\left(\left[G^{\prime}\right]\right)L2([G])=⨁χ∈X(G)Lχ2([G]),L2([G′])=⨁χ′∈X(G′)Lχ′2([G′])
The automorphic kernels K f , K f K f , K f ′ K_(f),K_(f^('))K_{f}, K_{f^{\prime}}Kf,Kf′ decompose accordingly into series K f = χ K f , χ K f = ∑ χ   K f , χ K_(f)=sum_(chi)K_(f,chi)K_{f}=\sum_{\chi} K_{f, \chi}Kf=∑χKf,χ, K f = χ K f , χ K f ′ = ∑ χ ′   K f ′ , χ ′ K_(f^('))=sum_(chi^('))K_(f^('),chi^('))K_{f^{\prime}}=\sum_{\chi^{\prime}} K_{f^{\prime}, \chi^{\prime}}Kf′=∑χ′Kf′,χ′ where K f , χ K f , χ K_(f,chi)K_{f, \chi}Kf,χ and K f , χ K f ′ , χ K_(f^('),chi)K_{f^{\prime}, \chi}Kf′,χ are kernel functions representing the restrictions R χ ( f ) R χ ( f ) R_(chi)(f)R_{\chi}(f)Rχ(f) and R χ ( f ) R χ ′ f ′ R_(chi^('))(f^('))R_{\chi^{\prime}}\left(f^{\prime}\right)Rχ′(f′) of the right convolution operators R ( f ) R ( f ) R(f)R(f)R(f) and R ( f ) R f ′ R(f^('))R\left(f^{\prime}\right)R(f′) to L χ 2 ( [ G ] ) L χ 2 ( [ G ] ) L_(chi)^(2)([G])L_{\chi}^{2}([G])Lχ2([G]) and L χ 2 ( [ G ] ) L χ ′ 2 G ′ L_(chi^('))^(2)([G^(')])L_{\chi^{\prime}}^{2}\left(\left[G^{\prime}\right]\right)Lχ′2([G′]), respectively. The distributions f J χ ( f ) f ↦ J χ ( f ) f|->J_(chi)(f)f \mapsto J_{\chi}(f)f↦Jχ(f) and f I χ ( f ) f ′ ↦ I χ ′ f ′ f^(')|->I_(chi^('))(f^('))f^{\prime} \mapsto I_{\chi^{\prime}}\left(f^{\prime}\right)f′↦Iχ′(f′) are then roughly defined by applying the same regularization procedure that Zydor used for the expressions J ( f ) J ( f ) J(f)J(f)J(f) and I ( f ) I f ′ I(f^('))I\left(f^{\prime}\right)I(f′) up to replacing the integrands by K f , χ K f , χ K_(f,chi)K_{f, \chi}Kf,χ and K f , χ K f ′ , χ ′ K_(f^('),chi^('))K_{f^{\prime}, \chi^{\prime}}Kf′,χ′ respectively, that is, in symbolic terms:
(4.5) J χ ( f ) = [ H ] × [ H ] r e g K f , χ ( h 1 , h 2 ) d h 1 d h 2 I χ ( f ) = [ H 1 ] × [ H 2 ] r e g K f , χ ( h 1 , h 2 ) η ( h 2 ) d h 1 d h 2 (4.5) J χ ( f ) = ∫ [ H ] × [ H ] r e g   K f , χ h 1 , h 2 d h 1 d h 2 I χ ′ f ′ = ∫ H 1 × H 2 r e g   K f ′ , χ ′ h 1 , h 2 η h 2 d h 1 d h 2 {:[(4.5)J_(chi)(f)=int_([H]xx[H])^(reg)K_(f,chi)(h_(1),h_(2))dh_(1)dh_(2)],[I_(chi^('))(f^('))=int_([H_(1)]xx[H_(2)])^(reg)K_(f^('),chi^('))(h_(1),h_(2))eta(h_(2))dh_(1)dh_(2)]:}\begin{align*} J_{\chi}(f) & =\int_{[H] \times[H]}^{\mathrm{reg}} K_{f, \chi}\left(h_{1}, h_{2}\right) d h_{1} d h_{2} \tag{4.5}\\ I_{\chi^{\prime}}\left(f^{\prime}\right) & =\int_{\left[H_{1}\right] \times\left[H_{2}\right]}^{\mathrm{reg}} K_{f^{\prime}, \chi^{\prime}}\left(h_{1}, h_{2}\right) \eta\left(h_{2}\right) d h_{1} d h_{2} \end{align*}(4.5)Jχ(f)=∫[H]×[H]regKf,χ(h1,h2)dh1dh2Iχ′(f′)=∫[H1]×[H2]regKf′,χ′(h1,h2)η(h2)dh1dh2
However, the expansions (4.4) are of little use as they stand and need to be suitably refined to allow for a meaningful comparison of the trace formulae. In Arthur's terminology, (4.4) are coarse spectral expansions and we need refined spectral expansions for each of the terms J χ ( f ) J χ ( f ) J_(chi)(f)J_{\chi}(f)Jχ(f) or I χ ( f ) I χ ′ f ′ I_(chi^('))(f^('))I_{\chi^{\prime}}\left(f^{\prime}\right)Iχ′(f′).
This problem has so far proved to be a very difficult for general cuspidal data χ χ chi\chiχ and χ χ ′ chi^(')\chi^{\prime}χ′. However, a recent result of mine in collaboration with Y. Liu, W. Zhang, and X. Zhu [17] allows isolating in the coarse spectral expansions (4.4) the only terms that are eventually of interest consequently reducing the problem to some very particular cuspidal data χ χ ′ chi^(')\chi^{\prime}χ′ of G G ′ G^(')G^{\prime}G′.
The result proved in [17] is very general so let us place ourself for one moment in the framework of an arbitrary connected reductive group G G GGG over the number field F F FFF. Let Σ Î£ Sigma\SigmaΣ be a set of non-Archimedean places of F F FFF (possibly infinite) such that for each v v ∈ v inv \inv∈ Σ Î£ Sigma\SigmaΣ, the group G v G v G_(v)G_{v}Gv is unramified and fix a hyperspecial compact subgroup K v G v K v ⊂ G v K_(v)subG_(v)K_{v} \subset G_{v}Kv⊂Gv with K v = G ( O v ) K v = G O v K_(v)=G(O_(v))K_{v}=G\left(\mathcal{O}_{v}\right)Kv=G(Ov) for almost all v Σ v ∈ Σ v in Sigmav \in \Sigmav∈Σ. We let X Σ ( G ) X Σ ( G ) X_(Sigma)(G)\mathcal{X}_{\Sigma}(G)XΣ(G) be the set of Σ Î£ Sigma\SigmaΣ-unramified cuspidal data of G G GGG, that is, the cuspidal data represented by pairs ( M , σ ) ( M , σ ) (M,sigma)(M, \sigma)(M,σ) with σ σ sigma\sigmaσ unramified at all places of v Σ v ∈ Σ v in Sigmav \in \Sigmav∈Σ (with respect to K v K v K_(v)K_{v}Kv or, rather, the hyperspecial subgroup it induces in M v M v M_(v)M_{v}Mv ). For χ X Σ ( G ) χ ∈ X Σ ( G ) chi inX_(Sigma)(G)\chi \in \mathcal{X}_{\Sigma}(G)χ∈XΣ(G), we define its Σ Î£ Sigma\SigmaΣ-near equivalence class, henceforth denoted by N Σ ( χ ) N Σ ( χ ) N_(Sigma)(chi)\mathcal{N}_{\Sigma}(\chi)NΣ(χ), as the set of all cuspidal data χ X Σ ( G ) χ ′ ∈ X Σ ( G ) chi^(')inX_(Sigma)(G)\chi^{\prime} \in \mathcal{X}_{\Sigma}(G)χ′∈XΣ(G) such that if χ χ chi\chiχ and χ χ ′ chi^(')\chi^{\prime}χ′ are represented by pairs ( M , σ ) ( M , σ ) (M,sigma)(M, \sigma)(M,σ) and ( M , σ ) M ′ , σ ′ (M^('),sigma^('))\left(M^{\prime}, \sigma^{\prime}\right)(M′,σ′) respectively, then there exist automorphic unramified characters λ λ lambda\lambdaλ and λ λ ′ lambda^(')\lambda^{\prime}λ′ of M ( A F ) M A F M(A_(F))M\left(\mathbb{A}_{F}\right)M(AF)
and M ( A F ) M ′ A F M^(')(A_(F))M^{\prime}\left(\mathbb{A}_{F}\right)M′(AF), respectively, with the property that for every v Σ v ∈ Σ v in Sigmav \in \Sigmav∈Σ the Satake parameters of the unique K v K v K_(v)K_{v}Kv-unramified subquotients in I P v G v ( σ v λ v ) I P v G v σ v ⊗ λ v I_(P_(v))^(G_(v))(sigma_(v)oxlambda_(v))I_{P_{v}}^{G_{v}}\left(\sigma_{v} \otimes \lambda_{v}\right)IPvGv(σv⊗λv) and I P v G v ( σ v λ v ) ( I P v ′ G v σ v ′ ⊗ λ v ′ I_(P_(v)^('))^(G_(v))(sigma_(v)^(')oxlambda_(v)^('))(:}I_{P_{v}^{\prime}}^{G_{v}}\left(\sigma_{v}^{\prime} \otimes \lambda_{v}^{\prime}\right)\left(\right.IPv′Gv(σv′⊗λv′)( where P , P P , P ′ P,P^(')P, P^{\prime}P,P′ are arbitrary chosen parabolics with Levi components M , M M , M ′ M,M^(')M, M^{\prime}M,M′ ) are isomorphic. We also fix a compact-open subgroup K = v S f K v K = ∏ v ∈ S f   K v K=prod_(v inS_(f))K_(v)K=\prod_{v \in S_{f}} K_{v}K=∏v∈SfKv of G ( A f ) G A f G(A_(f))G\left(\mathbb{A}_{f}\right)G(Af) (where S f S f S_(f)S_{f}Sf denotes the set of finite places of F F FFF and K v K v K_(v)K_{v}Kv coincides with the previous choice of hyperspecial subgroup when v Σ v ∈ Σ v in Sigmav \in \Sigmav∈Σ ) and we define the Schwartz space of K K KKK-biinvariant functions on G ( A F ) G A F G(A_(F))G\left(\mathbb{A}_{F}\right)G(AF) as the restricted tensor product
ς K ( G ( A F ) ) = S ( G ( F ) ) v S f C c ( K v G v / K v ) Ï‚ K G A F = S G F ∞ ⊗ ⨂ v ∈ S f ′   C c K v ∖ G v / K v Ï‚_(K)(G(A_(F)))=S(G(F_(oo)))ox⨂_(v inS_(f))^(')C_(c)(K_(v)\\G_(v)//K_(v))\varsigma_{K}\left(G\left(\mathbb{A}_{F}\right)\right)=\mathcal{S}\left(G\left(F_{\infty}\right)\right) \otimes \bigotimes_{v \in S_{f}}^{\prime} C_{c}\left(K_{v} \backslash G_{v} / K_{v}\right)Ï‚K(G(AF))=S(G(F∞))⊗⨂v∈Sf′Cc(Kv∖Gv/Kv)
where C c ( K v G v / K v ) C c K v ∖ G v / K v C_(c)(K_(v)\\G_(v)//K_(v))C_{c}\left(K_{v} \backslash G_{v} / K_{v}\right)Cc(Kv∖Gv/Kv) denotes the space of bi- K v K v K_(v)K_{v}Kv-invariant compactly supported functions on G v G v G_(v)G_{v}Gv (that is the K v K v K_(v)K_{v}Kv-spherical Hecke algebra when v Σ v ∈ Σ v in Sigmav \in \Sigmav∈Σ ), F F ∞ F_(oo)F_{\infty}F∞ is the product of the Archimedean completions of F F FFF and S ( G ( F ) ) S G F ∞ S(G(F_(oo)))S\left(G\left(F_{\infty}\right)\right)S(G(F∞)) stands for the Schwartz space of the reductive Lie group G ( F ) G F ∞ G(F_(oo))G\left(F_{\infty}\right)G(F∞) in the sense of [19]. More precisely, S ( G ( F ) ) S G F ∞ S(G(F_(oo)))\mathcal{S}\left(G\left(F_{\infty}\right)\right)S(G(F∞)) is the space of smooth functions f : G ( F ) C f : G F ∞ → C f:G(F_(oo))rarrCf: G\left(F_{\infty}\right) \rightarrow \mathbb{C}f:G(F∞)→C such that for every polynomial differential operator on G ( F ) G F ∞ G(F_(oo))G\left(F_{\infty}\right)G(F∞), the derivatives D f D f DfD fDf is bounded or, equivalently, such that for every left- (or right)invariant differential operator X , X f X , X f X,XfX, X fX,Xf is decreasing faster than the inverse of any polynomial on G ( F ) G F ∞ G(F_(oo))G\left(F_{\infty}\right)G(F∞).
The Schwartz space S ( G ( F ) ) S G F ∞ S(G(F_(oo)))S\left(G\left(F_{\infty}\right)\right)S(G(F∞)) is naturally a Fréchet algebra under the convolution product and we also set
M ( G ) = End cont , s ( G ( F ) ) bimod ( S ( G ( F ) ) ) M ∞ ( G ) = End cont  , s G F ∞ − bimod ⁡ S G F ∞ M_(oo)(G)=End_("cont ",s(G(F_(oo)))-bimod)(S(G(F_(oo))))\mathcal{M}_{\infty}(G)=\operatorname{End}_{\text {cont }, s\left(G\left(F_{\infty}\right)\right)-\operatorname{bimod}}\left(S\left(G\left(F_{\infty}\right)\right)\right)M∞(G)=Endcont ,s(G(F∞))−bimod⁡(S(G(F∞)))
for the space of continuous endomorphisms of S ( G ( F ) ) S G F ∞ S(G(F_(oo)))S\left(G\left(F_{\infty}\right)\right)S(G(F∞)) seen as a bimodule over itself. This is an algebra acting on any smooth admissible Fréchet representation of moderate growth of G ( F ) G F ∞ G(F_(oo))G\left(F_{\infty}\right)G(F∞) in the sense of Casselman-Wallach. Moreover, as an application of a form of Schur lemma, for every irreducible Casselman-Wallach representation π Ï€ ∞ pi_(oo)\pi_{\infty}π∞ of G ( F ) G F ∞ G(F_(oo))G\left(F_{\infty}\right)G(F∞) and every μ M ( G ) μ ∞ ∈ M ∞ ( G ) mu_(oo)inM_(oo)(G)\mu_{\infty} \in \mathcal{M}_{\infty}(G)μ∞∈M∞(G) there exists a scalar μ ( π ) C μ ∞ Ï€ ∞ ∈ C mu_(oo)(pi_(oo))inC\mu_{\infty}\left(\pi_{\infty}\right) \in \mathbb{C}μ∞(π∞)∈C such that π ( μ ) = μ ( π ) I d Ï€ ∞ μ ∞ = μ ∞ Ï€ ∞ I d pi_(oo)(mu_(oo))=mu_(oo)(pi_(oo))Id\pi_{\infty}\left(\mu_{\infty}\right)=\mu_{\infty}\left(\pi_{\infty}\right) I dπ∞(μ∞)=μ∞(π∞)Id. Thus, M ( G ) M ∞ ( G ) M_(oo)(G)\mathcal{M}_{\infty}(G)M∞(G) can be seen as some big algebra of multipliers for δ ( G ( F ) ) δ G F ∞ delta(G(F_(oo)))\delta\left(G\left(F_{\infty}\right)\right)δ(G(F∞)). We also define the algebra of Σ Î£ Sigma\SigmaΣ-multipliers as the restricted tensor product
M Σ ( G ) = M ( G ) v Σ H ( G v , K v ) M Σ ( G ) = M ∞ ( G ) ⨂ v ∈ Σ ′   H G v , K v M_(Sigma)(G)=M_(oo)(G)⨂_(v in Sigma)^(')H(G_(v),K_(v))\mathcal{M}_{\Sigma}(G)=\mathcal{M}_{\infty}(G) \bigotimes_{v \in \Sigma}^{\prime} \mathscr{H}\left(G_{v}, K_{v}\right)MΣ(G)=M∞(G)⨂v∈Σ′H(Gv,Kv)
where, for v Σ , H ( G v , K v ) = C c ( K v G v / K v ) v ∈ Σ , H G v , K v = C c K v ∖ G v / K v v in Sigma,H(G_(v),K_(v))=C_(c)(K_(v)\\G_(v)//K_(v))v \in \Sigma, \mathscr{H}\left(G_{v}, K_{v}\right)=C_{c}\left(K_{v} \backslash G_{v} / K_{v}\right)v∈Σ,H(Gv,Kv)=Cc(Kv∖Gv/Kv) is the spherical Hecke algebra. Then, M Σ ( G ) M Σ ( G ) M_(Sigma)(G)\mathcal{M}_{\Sigma}(G)MΣ(G) acts naturally on the global Schwartz space ς K ( G ( A F ) ) Ï‚ K G A F Ï‚_(K)(G(A_(F)))\varsigma_{K}\left(G\left(\mathbb{A}_{F}\right)\right)Ï‚K(G(AF)), and we shall denote this action as the convolution product ∗ ***∗. One of the main result of [17] can now be stated as follows:
Theorem 4.2 (Beuzart-Plessis-Liu-Zhang-Zhu). Let χ X Σ ( G ) χ ∈ X Σ ( G ) chi inX_(Sigma)(G)\chi \in \mathcal{X}_{\Sigma}(G)χ∈XΣ(G). Then, there exists a multiplier μ χ M Σ ( G ) μ χ ∈ M Σ ( G ) mu_(chi)inM_(Sigma)(G)\mu_{\chi} \in \mathcal{M}_{\Sigma}(G)μχ∈MΣ(G) such that for every Schwartz function f S K ( G ( A F ) ) f ∈ S K G A F f inS_(K)(G(A_(F)))f \in S_{K}\left(G\left(\mathbb{A}_{F}\right)\right)f∈SK(G(AF)) and every other cuspidal datum χ X Σ ( G ) χ ′ ∈ X Σ ( G ) chi^(')inX_(Sigma)(G)\chi^{\prime} \in \mathcal{X}_{\Sigma}(G)χ′∈XΣ(G), we have
R χ ( μ χ f ) = { R χ ( f ) if χ N Σ ( χ ) 0 otherwise R χ ′ μ χ ∗ f = R χ ′ ( f )  if  χ ′ ∈ N Σ ( χ ) 0  otherwise  R_(chi^('))(mu_(chi)**f)={[R_(chi^('))(f)," if "chi^(')inN_(Sigma)(chi)],[0," otherwise "]:}R_{\chi^{\prime}}\left(\mu_{\chi} * f\right)= \begin{cases}R_{\chi^{\prime}}(f) & \text { if } \chi^{\prime} \in \mathcal{N}_{\Sigma}(\chi) \\ 0 & \text { otherwise }\end{cases}Rχ′(μχ∗f)={Rχ′(f) if χ′∈NΣ(χ)0 otherwise 
The above theorem can be roughly paraphrased by saying that the multiplier μ χ μ χ mu_(chi)\mu_{\chi}μχ "isolates" the near-equivalence class N Σ ( χ ) N Σ ( χ ) N_(Sigma)(chi)\mathcal{N}_{\Sigma}(\chi)NΣ(χ) from the other cuspidal data. A large part of the proof given in [17] consists in establishing the existence of a large subalgebra of M ( G ) M ∞ ( G ) M_(oo)(G)\mathcal{M}_{\infty}(G)M∞(G) which admits an explicit spectral description, that is, through its action on irreducible Casselman-Wallach representations of G ( F ) G F ∞ G(F_(oo))G\left(F_{\infty}\right)G(F∞). The algebra thus constructed generalizes Arthur's multipliers [5] and, moreover, builds on previous work of Delorme [21].
Going back to the setting of the Jacquet-Rallis trace formulae, the above theorem can be applied to isolate in the expansions (4.4) the automorphic L L LLL-packet of a given cuspidal automorphic representation π Ï€ pi\piÏ€ of G ( A F ) G A F G(A_(F))G\left(\mathbb{A}_{F}\right)G(AF), on the one hand, and the cuspidal datum χ χ chi\chiχ of G G ′ G^(')G^{\prime}G′ "supporting" its base-change π E Ï€ E pi_(E)\pi_{E}Ï€E, on the other hand. Moreover, essentially using the spectral characterization of Theorem 3.2 for the transfer, this can be done by multipliers μ π M Σ ( G ) μ Ï€ ∈ M Σ ( G ) mu_(pi)inM_(Sigma)(G)\mu_{\pi} \in \mathcal{M}_{\Sigma}(G)μπ∈MΣ(G) and μ χ M Σ ( G ) μ χ ∈ M Σ G ′ mu_(chi)inM_(Sigma)(G^('))\mu_{\chi} \in \mathcal{M}_{\Sigma}\left(G^{\prime}\right)μχ∈MΣ(G′) that are compatible with the Jacquet-Rallis transfer in the following sense: if f = v f v S K ( G ( A F ) ) f = ∏ v   f v ∈ S K G A F f=prod_(v)f_(v)inS_(K)(G(A_(F)))f=\prod_{v} f_{v} \in S_{K}\left(G\left(\mathbb{A}_{F}\right)\right)f=∏vfv∈SK(G(AF)) and f = v f v S K ( G ( A F ) ) f ′ = ∏ v   f v ′ ∈ S K ′ G ′ A F f^(')=prod_(v)f_(v)^(')inS_(K^('))(G^(')(A_(F)))f^{\prime}=\prod_{v} f_{v}^{\prime} \in S_{K^{\prime}}\left(G^{\prime}\left(\mathbb{A}_{F}\right)\right)f′=∏vfv′∈SK′(G′(AF)) are transfers of each other then so are μ π f μ Ï€ ∗ f mu_(pi)**f\mu_{\pi} * fμπ∗f and μ χ f μ χ ∗ f ′ mu_(chi)**f^(')\mu_{\chi} * f^{\prime}μχ∗f′ (where here we take Σ Î£ Sigma\SigmaΣ to consist of almost all places that split in E E EEE and for K , K K , K ′ K,K^(')K, K^{\prime}K,K′ arbitrary compact-open subgroups of G ( A f ) , G ( A f ) G A f , G ′ A f G(A_(f)),G^(')(A_(f))G\left(\mathbb{A}_{f}\right), G^{\prime}\left(\mathbb{A}_{f}\right)G(Af),G′(Af) that are hyperspecial at places in Σ Î£ Sigma\SigmaΣ ). All in all, applying these multipliers to global test functions f f fff and f f ′ f^(')f^{\prime}f′ that are transfers of each other, and comparing the geometric expansions (4.1), we obtain an identity of the following shape:
W π A cusp ( G W ) π E = π E J π ( f ) = I χ ( f ) ∑ W ′   ∑ Ï€ ′ ↪ A cusp  G W ′ Ï€ E ′ = Ï€ E   J Ï€ ′ ( f ) = I χ f ′ sum_(W^('))sum_({:[pi^(')↪A_("cusp ")(G^(W^(')))],[pi_(E)^(')=pi_(E)]:})J_(pi^('))(f)=I_(chi)(f^('))\sum_{W^{\prime}} \sum_{\substack{\pi^{\prime} \hookrightarrow \mathcal{A}_{\text {cusp }}\left(G^{W^{\prime}}\right) \\ \pi_{E}^{\prime}=\pi_{E}}} J_{\pi^{\prime}}(f)=I_{\chi}\left(f^{\prime}\right)∑W′∑π′↪Acusp (GW′)Ï€E′=Ï€EJπ′(f)=Iχ(f′)
where the outside left sum runs over isomorphism classes of Hermitian spaces of the same dimension as W W WWW (or, equivalently, relevant pure inner forms of G G GGG ). Besides, as a consequence of the local Gan-Gross-Prasad conjecture, when π E Ï€ E pi_(E)\pi_{E}Ï€E is generic, the left-hand side contains at most one nonzero term. Thus, as a final step to establish the Gan-Gross-Prasad and IchinoIkeda conjectures, it only remains to analyze the distribution I χ I χ I_(chi)I_{\chi}Iχ. When the base-change π E Ï€ E pi_(E)\pi_{E}Ï€E is itself cuspidal, that is, when χ = { ( G , π E ) } χ = G ′ , Ï€ E chi={(G^('),pi_(E))}\chi=\left\{\left(G^{\prime}, \pi_{E}\right)\right\}χ={(G′,Ï€E)}, by the works of Jacquet-PiatetskiShapiro-Shalika and Flicker-Rallis already recalled, I χ I χ I_(chi)I_{\chi}Iχ essentially factors as the product of the local relative characters I π E , v I Ï€ E , v I_(pi_(E,v))I_{\pi_{E, v}}IÏ€E,v and Theorem 3.2 then allows to conclude. However, in general a similar factorization of I χ I χ I_(chi)I_{\chi}Iχ is far from obvious and was actually established in my joint work with Chaudouard and Zydor [16]. It is exactly of the shape predicted by the Ichino-Ikeda conjecture. More precisely:
Theorem 4.3 (Beuzart-Plessis-Chaudouard-Zydor). Let π Ï€ pi\piÏ€ be a cuspidal automorphic representation of G ( A F ) G A F G(A_(F))G\left(\mathbb{A}_{F}\right)G(AF) whose base-change π E Ï€ E pi_(E)\pi_{E}Ï€E is generic. Let χ χ chi\chiχ be the cuspidal datum of G G ′ G^(')G^{\prime}G′ such that π E Ï€ E pi_(E)\pi_{E}Ï€E contributes to the spectral decomposition of L χ 2 ( [ G ] ) L χ 2 G ′ L_(chi)^(2)([G^(')])L_{\chi}^{2}\left(\left[G^{\prime}\right]\right)Lχ2([G′]). Then, for every factor izable test function f = v f v S ( G ( A F ) ) f ′ = ∏ v   f v ′ ∈ S G ′ A F f^(')=prod_(v)f_(v)^(')in S(G^(')(A_(F)))f^{\prime}=\prod_{v} f_{v}^{\prime} \in S\left(G^{\prime}\left(\mathbb{A}_{F}\right)\right)f′=∏vfv′∈S(G′(AF)), we have
(4.6) I χ ( f ) = 1 | S π | v I π E , v ( f v ) (4.6) I χ f ′ = 1 S Ï€ ∏ v ′   I Ï€ E , v f v ′ {:(4.6)I_(chi)(f^('))=(1)/(|S_(pi)|)prod_(v)^(')I_(pi_(E,v))(f_(v)^(')):}\begin{equation*} I_{\chi}\left(f^{\prime}\right)=\frac{1}{\left|S_{\pi}\right|} \prod_{v}^{\prime} I_{\pi_{E, v}}\left(f_{v}^{\prime}\right) \tag{4.6} \end{equation*}(4.6)Iχ(f′)=1|SÏ€|∏v′IÏ€E,v(fv′)
where the product has to be understood, as for (2.4), "in the sense of L-functions."
In [16], two proofs are actually given of the above theorem: one using truncations operators and the other one based on the global theory of Zeta integrals. For both methods, a crucial step is to spectrally expand the restriction of the Flicker-Rallis period (that is, the integral over [ H 2 ] H 2 [H_(2)]\left[H_{2}\right][H2] ) to functions φ L χ 2 ( [ G ] ) φ ∈ L χ 2 G ′ varphi inL_(chi)^(2)([G^(')])\varphi \in L_{\chi}^{2}\left(\left[G^{\prime}\right]\right)φ∈Lχ2([G′]) that are sufficiently rapidly decreasing. A consequence of this computation is that this period only depends on the π E Ï€ E pi_(E)\pi_{E}Ï€E-component of φ φ varphi\varphiφ and it is mainly for this reason that the contribution of χ χ chi\chiχ to the Jacquet-Rallis trace formula I ( f ) I f ′ I(f^('))I\left(f^{\prime}\right)I(f′) is eventually discrete (although in the case at hand, L χ 2 ( [ G ] ) L χ 2 G ′ L_(chi)^(2)([G^(')])L_{\chi}^{2}\left(\left[G^{\prime}\right]\right)Lχ2([G′]) usually has a purely continuous spectrum). For this, the truncation method is based on the work of Jacquet-Lapid-Rogawski who have defined and studied generalizations of Arthur's truncation operator to the setting of Galois periods and proved analogs of the Maass-Selberg relations in this context. On the other hand, the other method starts by expressing the Flicker-Rallis period as a residue of the integral over [ H 2 ] H 2 [H_(2)]\left[\mathrm{H}_{2}\right][H2] of φ φ varphi\varphiφ against an Eisenstein series. Unfolding carefully this expression as in the work of Flicker-Rallis, we can rewrite it as a Zeta integral of the sort that represents Asai L L LLL-functions. The precise location of the poles of these L L LLL-functions, as well as an explicit residue computation of a family of distributions, then allows to conclude.
Finally, let me mention that in work in progress with P.-H. Chaudouard, we are able to analyze the contributions to the Jacquet-Rallis trace formula of more general cuspidal data χ X ( G ) χ ∈ X G ′ chi inX(G^('))\chi \in \mathcal{X}\left(G^{\prime}\right)χ∈X(G′) than that appearing in Theorem [16]. The final result is similar to (4.6) except that the right-hand side has to be integrated over a certain family of automorphic representations π Ï€ pi\piÏ€ of G ( A F ) G A F G(A_(F))G\left(\mathbb{A}_{F}\right)G(AF). More precisely, our results include some cuspidal data supporting the basechanges of automorphic representations of G = U ( V ) × U ( W ) G = U ( V ) × U ( W ) G=U(V)xx U(W)G=U(V) \times U(W)G=U(V)×U(W) that are Eisenstein in the first factor and cuspidal in the second. In this particular case, the contribution of the corresponding cuspidal datum to the trace formula J ( f ) J ( f ) J(f)J(f)J(f) is absolutely convergent and a refined spectral expansion can readily be obtained as an integral of Gan-Gross-Prasad periods between a cusp form and an Eisenstein series. These last periods are related, by some unfolding, to Bessel periods of cusp forms on smaller unitary groups. For this reason, our extension of Theorem 4.3 with Chaudouard should have similar applications to the Gan-Gross-Prasad and Ichino-Ikeda conjectures for general Bessel periods.

5. LOOKING FORWARD

As illustrated in the previous sections, various trace formula approaches to the GanGross-Prasad conjectures for unitary groups have been very successful. However, despite these favorable and definite results, these developments also raise interesting questions or have lead to fertile new research direction:
  • First, there is the question of whether similar techniques can be applied to prove the global Gan-Gross-Prasad conjectures for other groups. Indeed, the original conjectures in [23] also include general Bessel periods on (product of) orthogonal groups SO ( n ) × SO ( m ) ( n m [ 2 ] ) SO ⁡ ( n ) × SO ⁡ ( m ) ( n ≢ m [ 2 ] ) SO(n)xx SO(m)(n≢m[2])\operatorname{SO}(n) \times \operatorname{SO}(m)(n \not \equiv m[2])SO⁡(n)×SO⁡(m)(n≢m[2]), as well as so-called Fourier-Jacobi periods on unitary groups U ( n ) × U ( m ) ( n m U ( n ) × U ( m ) ( n ≡ m U(n)xx U(m)(n-=mU(n) \times U(m)(n \equiv mU(n)×U(m)(n≡m [2]) or symplectic/metaplectic groups Mp ( n ) × Sp ( m ) Mp ⁡ ( n ) × Sp ⁡ ( m ) Mp(n)xx Sp(m)\operatorname{Mp}(n) \times \operatorname{Sp}(m)Mp⁡(n)×Sp⁡(m). In the case of U ( n ) × U ( n ) U ( n ) × U ( n ) U(n)xx U(n)U(n) \times U(n)U(n)×U(n), a relative trace formula approach
    has been proposed by Y. Liu and further developed by H. Xue [52]. However, the situation is not as complete as for the Jacquet-Rallis trace formulae in the case of U ( n + 1 ) × U ( n ) U ( n + 1 ) × U ( n ) U(n+1)xx U(n)U(n+1) \times U(n)U(n+1)×U(n). It would be interesting to see if the latest developments, in particular those from my joint work with Chaudouard and Zydor [16], can be adapted to this setting. This could possibly lead to a proof of the GanGross-Prasad conjecture for general Fourier-Jacobi periods on unitary groups. The situation for orthogonal and symplectic/metaplectic groups is much less satisfactory and there is no clear approach through a comparison of relative trace formulae, yet. This is due in particular to the fact that, instead of the FlickerRallis periods, in these cases we are naturally lead to consider period integrals originally studied by Bump-Ginzburg that detect cuspidal automorphic representations of G L ( n ) G L ( n ) GL(n)\mathrm{GL}(n)GL(n) of orthogonal type. These period integrals involve the product of two exceptional theta series on a double cover of GL ( n ) GL ⁡ ( n ) GL(n)\operatorname{GL}(n)GL⁡(n) and do not have any obvious geometric realizations (except when n = 2 n = 2 n=2n=2n=2 ). This makes the task of writing a geometric expansion for the corresponding trace formulae quite unclear. It would certainly be interesting to see if the recent Hamiltonian duality picture of Ben Zvi-Sakellaridis-Venkatesh can shed some light on this matter (in particular, by associating a Hamiltonian space to the Bump-Ginzburg periods).
  • In the local setting, the new trace formulae first discovered by Waldspurger [47] and further developed in [12] seem to be of quite broad applicability to all kind of distinction problems. Actually, similar trace formulae have already been established in other contexts [ 11 , 18 , 50 ] [ 11 , 18 , 50 ] [11,18,50][11,18,50][11,18,50] with new applications in the spirit of "relative Langlands functorialities" each time. However, all these developments have been made on a case-by-case basis so far and it would be very interesting and instructive to elaborate a general theory. In particular, in view of the proposal by Sakellaridis-Venkatesh [41] of a general framework for the relative Langlands program, we could hope to establish general local relative trace formulae for the L 2 L 2 L^(2)L^{2}L2 spaces of spherical varieties X X XXX and relate those to the dual group construction of Sakellaridis-Venkatesh.
  • Finally, in a slightly different direction the general isolation Theorem 4.2 clearly has the potential to be applied in other context, e.g., it would be interesting to see if it can be used as a technical device to simplify some other known comparison of trace formulae. Another intriguing question is to look for a precise spectral description of the (abstract) multiplier algebra M ( G ) M ∞ ( G ) M_(oo)(G)\mathcal{M}_{\infty}(G)M∞(G) and in [17], we actually argue that M ( G ) M ∞ ( G ) M_(oo)(G)\mathcal{M}_{\infty}(G)M∞(G) should be seen as the natural Archimedean analog of the Bernstein center for p p ppp-adic groups.

ACKNOWLEDGMENTS

First and foremost, I thank my former PhD advisor Jean-Loup Waldspurger for introducing me to this fertile circle of ideas and for always being so generous in sharing his insights on these problems.
I am also grateful to my colleagues and coauthors Pierre-Henri Chaudouard, Patrick Delorme, Wee-Teck Gan, Yifeng Liu, Dipendra Prasad, Yiannis Sakellaridis, Chen Wan, Wei Zhang, Xinwen Zhu, and Michal Zydor for many inspiring discussions and exchanges over the years.

FUNDING

The author's work was partially supported by the Excellence Initiative of Aix-Marseille University (A*MIDEX), a French "Investissements d'Avenir" program.

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RAPHAËL BEUZART-PLESSIS

Aix Marseille Univ, CNRS, I2M, Marseille, France, raphael.beuzart-plessis@univ-amu.fr

THE COHOMOLOGY OF SHIMURA VARIETIES WITH TORSION COEFFICIENTS

ANA CARAIANI

ABSTRACT

In this article, we survey recent work on some vanishing conjectures for the cohomology of Shimura varieties with torsion coefficients, under both local and global conditions. We discuss the p p ppp-adic geometry of Shimura varieties and of the associated Hodge-Tate period morphism, and explain how this can be used to make progress on these conjectures. Finally, we describe some applications of these results, in particular to the proof of the Sato-Tate conjecture for elliptic curves over CM fields.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11G18; Secondary 11R39, 11F33, 11F70, 11F75, 11F80

KEYWORDS

Shimura varieties, locally symmetric spaces, automorphic representations, Galois representations, p p ppp-adic Langlands programme

1. INTRODUCTION

Shimura varieties are algebraic varieties defined over number fields and equipped with many symmetries, which often provide a geometric realization of the Langlands correspondence. After base change to C C C\mathbb{C}C, they are closely related to certain locally symmetric spaces, but the beauty of Shimura varieties lies in their rich arithmetic.
To describe a Shimura variety, one needs to start with a Shimura datum ( G , X ) ( G , X ) (G,X)(G, X)(G,X). Here, G G GGG is a connected reductive group over Q Q Q\mathbb{Q}Q and X X XXX is a conjugacy class of homomorphisms h : Res C / R G m G R h : Res C / R ⁡ G m → G R h:Res_(C//R)G_(m)rarrG_(R)h: \operatorname{Res}_{\mathbb{C} / \mathbb{R}} \mathbb{G}_{m} \rightarrow G_{\mathbb{R}}h:ResC/R⁡Gm→GR of algebraic groups over R R R\mathbb{R}R. Both G G GGG and X X XXX are required to satisfy certain highly restrictive axioms, cf. [22, §2.1]. In particular, this allows one to give the conjugacy class X X XXX a more geometric flavor, as a variation of polarisable Hodge structures. One can show that such an X X XXX is a disjoint union of finitely many copies of Hermitian symmetric domains.
Let K G ( A f ) K ⊂ G A f K sub G(A_(f))K \subset G\left(\mathbb{A}_{f}\right)K⊂G(Af) be a sufficiently small compact open subgroup (the precise technical condition is called "neat"). The double quotient G ( Q ) X × G ( A f ) / K G ( Q ) ∖ X × G A f / K G(Q)\\X xx G(A_(f))//KG(\mathbb{Q}) \backslash X \times G\left(\mathbb{A}_{f}\right) / KG(Q)∖X×G(Af)/K, a priori a complex manifold, comes from an algebraic variety S K S K S_(K)S_{K}SK defined over a number field E E EEE, called the reflex field of the Shimura datum. The varieties S K S K S_(K)S_{K}SK are smooth and quasiprojective. Their étale cohomology groups (with or without compact support) H ( c ) ( S K × E Q ¯ , Q ) H ( c ) ∗ S K × E Q ¯ , Q â„“ H_((c))^(**)(S_(K)xx_(E) bar(Q),Q_(â„“))H_{(c)}^{*}\left(S_{K} \times_{E} \overline{\mathbb{Q}}, \mathbb{Q}_{\ell}\right)H(c)∗(SK×EQ¯,Qâ„“) are equipped with two kinds of symmetries. There is a Hecke symmetry coming from varying the level, i.e., the compact open subgroup K K KKK, and considering various transition morphisms between Shimura varieties at different levels. There is also a Galois symmetry, coming from the natural action of Gal ( E ¯ / E ) Gal ⁡ ( E ¯ / E ) Gal( bar(E)//E)\operatorname{Gal}(\bar{E} / E)Gal⁡(E¯/E) on étale cohomology.
For this reason, Shimura varieties have played an important role in realizing instances of the global Langlands correspondence over number fields. Indeed, a famous conjecture of Kottwitz predicts the relationship between the Galois representations occurring in the ℓ ℓ\ellℓ-adic étale cohomology of the Shimura varieties for G G GGG and those Galois representations associated with (regular, C C CCC-algebraic) cuspidal automorphic representations of G G GGG. See [64, REMARK 1.1.1] for a modern formulation of this conjecture.
There is a complete classification of groups that admit a Shimura datum. For example, if G = G S p 2 n G = G S p 2 n G=GSp_(2n)G=\mathrm{GSp}_{2 n}G=GSp2n, one can take X X XXX to be the Siegel double space
(1.1) { Z M n ( C ) Z = Z t , Im ( Z ) positive or negative definite } (1.1) Z ∈ M n ( C ) ∣ Z = Z t , Im ⁡ ( Z )  positive or negative definite  {:(1.1){Z inM_(n)(C)∣Z=Z^(t),Im(Z)" positive or negative definite "}:}\begin{equation*} \left\{Z \in \mathrm{M}_{n}(\mathbb{C}) \mid Z=Z^{t}, \operatorname{Im}(Z) \text { positive or negative definite }\right\} \tag{1.1} \end{equation*}(1.1){Z∈Mn(C)∣Z=Zt,Im⁡(Z) positive or negative definite }
The associated Shimura varieties are called Siegel modular varieties and they are moduli spaces of principally polarized abelian varieties. Many other Shimura varieties - those of socalled "abelian type" - can be studied using moduli-theoretic techniques, by relating them to Siegel modular varieties. See [39] for an excellent introduction to the subject, which is focused on examples.
In this article, we will be primarily concerned with the geometry of the Shimura varieties S K S K S_(K)S_{K}SK, after base change to a p p ppp-adic field, as well as with their étale cohomology groups H ( c ) ( S K × E Q ¯ , F ) H ( c ) ∗ S K × E Q ¯ , F â„“ H_((c))^(**)(S_(K)xx_(E) bar(Q),F_(â„“))H_{(c)}^{*}\left(S_{K} \times_{E} \overline{\mathbb{Q}}, \mathbb{F}_{\ell}\right)H(c)∗(SK×EQ¯,Fâ„“) with torsion coefficients. These groups are much less understood than their characteristic zero counterparts. We discuss certain conjectures about when these cohomology groups are expected to vanish, under both global and local conditions. Furthermore, we explain how the geometry of the Hodge-Tate period morphism, introduced in [53] and
refined in [17], can be used to make progress on these conjectures. Finally, we describe some applications of these results, in particular to the proof of the Sato-Tate conjecture for elliptic curves over CM fields [1].

2. A VANISHING CONJECTURE FOR LOCALLY SYMMETRIC SPACES

Let G / Q G / Q G//QG / \mathbb{Q}G/Q be a connected reductive group. We consider the symmetric space associated with the Lie group G ( R ) G ( R ) G(R)G(\mathbb{R})G(R), which we define as X = G ( R ) / K A X = G ( R ) / K ∞ ∘ A ∞ ∘ X=G(R)//K_(oo)^(@)A_(oo)^(@)X=G(\mathbb{R}) / K_{\infty}^{\circ} A_{\infty}^{\circ}X=G(R)/K∞∘A∞∘. Here, K K ∞ ∘ K_(oo)^(@)K_{\infty}^{\circ}K∞∘ is the connected component of the identity in a maximal compact subgroup K G ( R ) K ∞ ⊂ G ( R ) K_(oo)sub G(R)K_{\infty} \subset G(\mathbb{R})K∞⊂G(R), and A A ∞ ∘ A_(oo)^(@)A_{\infty}^{\circ}A∞∘ is the connected component of the identity inside the real points of the maximal Q Q Q\mathbb{Q}Q-split torus in the center of G G GGG. Given a neat compact open subgroup K G ( A f ) K ⊂ G A f K sub G(A_(f))K \subset G\left(\mathbb{A}_{f}\right)K⊂G(Af), we can form the double quotient X K = G ( Q ) X × G ( A f ) / K X K = G ( Q ) ∖ X × G A f / K X_(K)=G(Q)\\X xx G(A_(f))//KX_{K}=G(\mathbb{Q}) \backslash X \times G\left(\mathbb{A}_{f}\right) / KXK=G(Q)∖X×G(Af)/K, which we call a locally symmetric space for G G GGG. This is a smooth Riemannian manifold, which does not have a complex structure, in general.
Example 2.1. If G = S L 2 / Q G = S L 2 / Q G=SL_(2)//QG=\mathrm{SL}_{2} / \mathbb{Q}G=SL2/Q, we can identify X = S L 2 ( R ) / S O 2 ( R ) X = S L 2 ( R ) / S O 2 ( R ) X=SL_(2)(R)//SO_(2)(R)X=\mathrm{SL}_{2}(\mathbb{R}) / \mathrm{SO}_{2}(\mathbb{R})X=SL2(R)/SO2(R) with the upper halfplane H 2 = { z C Im z > 0 } H 2 = { z ∈ C ∣ Im ⁡ z > 0 } H^(2)={z inC∣Im z > 0}\mathbb{H}^{2}=\{z \in \mathbb{C} \mid \operatorname{Im} z>0\}H2={z∈C∣Im⁡z>0} equipped with the hyperbolic metric, on which S L 2 ( R ) S L 2 ( R ) SL_(2)(R)\mathrm{SL}_{2}(\mathbb{R})SL2(R) acts transitively by the isometries
z a z + b c z + d for z H 2 and ( a b c d ) S L 2 ( R ) z ↦ a z + b c z + d  for  z ∈ H 2  and  a b c d ∈ S L 2 ( R ) z|->(az+b)/(cz+d)" for "z inH^(2)quad" and "([a,b],[c,d])inSL_(2)(R)z \mapsto \frac{a z+b}{c z+d} \text { for } z \in \mathbb{H}^{2} \quad \text { and }\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \in \mathrm{SL}_{2}(\mathbb{R})z↦az+bcz+d for z∈H2 and (abcd)∈SL2(R)
Under this action, S O 2 ( R ) S O 2 ( R ) SO_(2)(R)\mathrm{SO}_{2}(\mathbb{R})SO2(R) is the stabilizer of the point i i iii. By strong approximation for S L 2 / Q S L 2 / Q SL_(2)//Q\mathrm{SL}_{2} / \mathbb{Q}SL2/Q, for any compact open subgroup K S L 2 ( Z ^ ) K ⊆ S L 2 ( Z ^ ) K subeSL_(2)( widehat(Z))K \subseteq \mathrm{SL}_{2}(\widehat{\mathbb{Z}})K⊆SL2(Z^), there is only one double coset S L 2 ( Q ) S L 2 ( A f ) / K S L 2 ( Q ) ∖ S L 2 A f / K SL_(2)(Q)\\SL_(2)(A_(f))//K\mathrm{SL}_{2}(\mathbb{Q}) \backslash \mathrm{SL}_{2}\left(\mathbb{A}_{f}\right) / KSL2(Q)∖SL2(Af)/K. Write Γ = S L 2 ( Q ) K Γ = S L 2 ( Q ) ∩ K Gamma=SL_(2)(Q)nn K\Gamma=\mathrm{SL}_{2}(\mathbb{Q}) \cap KΓ=SL2(Q)∩K, which will be a congruence subgroup contained in S L 2 ( Z ) S L 2 ( Z ) SL_(2)(Z)\mathrm{SL}_{2}(\mathbb{Z})SL2(Z). The locally symmetric spaces X K X K X_(K)X_{K}XK can be identified with quotients Γ H 2 Γ ∖ H 2 Gamma\\H^(2)\Gamma \backslash \mathbb{H}^{2}Γ∖H2. For Γ Î“ Gamma\GammaΓ neat, these quotients inherit the complex structure on H 2 H 2 H^(2)\mathbb{H}^{2}H2 and can be viewed as Riemann surfaces. Even more, these quotients arise from algebraic curves called modular curves, which are defined over finite extensions of Q Q Q\mathbb{Q}Q. Modular curves are examples of (connected) Shimura varieties. They represent moduli problems of elliptic curves endowed with additional structures. Even though they are some of the simplest Shimura varieties (the main complication being that they are noncompact), their geometry is already fascinating.
However, let F / Q F / Q F//QF / \mathbb{Q}F/Q be an imaginary quadratic field and take G = Res F / Q S L 2 G = Res F / Q ⁡ S L 2 G=Res_(F//Q)SL_(2)G=\operatorname{Res}_{F / \mathbb{Q}} \mathrm{SL}_{2}G=ResF/Q⁡SL2. Then we can identify the symmetric space X = S L 2 ( C ) / S U 2 ( R ) X = S L 2 ( C ) / S U 2 ( R ) X=SL_(2)(C)//SU_(2)(R)X=\mathrm{SL}_{2}(\mathbb{C}) / \mathrm{SU}_{2}(\mathbb{R})X=SL2(C)/SU2(R) with 3-dimensional hyperbolic space H 3 H 3 H^(3)\mathbb{H}^{3}H3. Once again, we can identify the locally symmetric spaces X K X K X_(K)X_{K}XK with quotients Γ H 3 Γ ∖ H 3 Gamma\\H^(3)\Gamma \backslash \mathbb{H}^{3}Γ∖H3, where Γ = S L 2 ( F ) K Γ = S L 2 ( F ) ∩ K Gamma=SL_(2)(F)nn K\Gamma=\mathrm{SL}_{2}(F) \cap KΓ=SL2(F)∩K is a congruence subgroup. In this case, the locally symmetric spaces are arithmetic hyperbolic 3-manifolds and do not admit a complex structure. In particular, we cannot speak of Shimura varieties in this setting.
In general, Shimura varieties are closely related to locally symmetric spaces, as in the first example, though the latter are much more general objects. For example, the locally symmetric spaces for G = Res F / Q G L n G = Res F / Q ⁡ G L n G=Res_(F//Q)GL_(n)G=\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{n}G=ResF/Q⁡GLn do not arise from Shimura varieties if n 3 n ≥ 3 n >= 3n \geq 3n≥3, and, for n = 2 n = 2 n=2n=2n=2, they can only be related to Shimura varieties if F F FFF is a totally real field. In some instances, such as for Res F / Q G L 2 Res F / Q ⁡ G L 2 Res_(F//Q)GL_(2)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{2}ResF/Q⁡GL2 with F F FFF a totally real field, one needs to replace
G ( R ) / K A G ( R ) / K ∞ ∘ A ∞ ∘ G(R)//K_(oo)^(@)A_(oo)^(@)G(\mathbb{R}) / K_{\infty}^{\circ} A_{\infty}^{\circ}G(R)/K∞∘A∞∘ by a slightly different quotient in order to obtain Shimura varieties. 1 1 ^(1){ }^{1}1 We now define the invariants
l 0 = rk ( G ( R ) ) rk ( K ) rk ( A ) and q 0 = 1 2 ( dim R ( X ) l 0 ) l 0 = rk ⁡ ( G ( R ) ) − rk ⁡ K ∞ − rk ⁡ A ∞  and  q 0 = 1 2 dim R ⁡ ( X ) − l 0 l_(0)=rk(G(R))-rk(K_(oo))-rk(A_(oo))quad" and "quadq_(0)=(1)/(2)(dim_(R)(X)-l_(0))l_{0}=\operatorname{rk}(G(\mathbb{R}))-\operatorname{rk}\left(K_{\infty}\right)-\operatorname{rk}\left(A_{\infty}\right) \quad \text { and } \quad q_{0}=\frac{1}{2}\left(\operatorname{dim}_{\mathbb{R}}(X)-l_{0}\right)l0=rk⁡(G(R))−rk⁡(K∞)−rk⁡(A∞) and q0=12(dimR⁡(X)−l0)
These were first introduced by Borel-Wallach in [5]. There, they show up naturally in the computation of the ( g , K g , K ∞ g,K_(oo)\mathrm{g}, K_{\infty}g,K∞ )-cohomology of tempered representations of G ( R ) G ( R ) G(R)G(\mathbb{R})G(R). In the Shimura variety setting, we consider the variants l 0 = l 0 ( G ad ) l 0 = l 0 G ad  l_(0)=l_(0)(G^("ad "))l_{0}=l_{0}\left(G^{\text {ad }}\right)l0=l0(Gad ) and q 0 = q 0 ( G ad ) q 0 = q 0 G ad  q_(0)=q_(0)(G^("ad "))q_{0}=q_{0}\left(G^{\text {ad }}\right)q0=q0(Gad ) because of the different quotient used. In this case, l 0 ( G ad ) l 0 G ad  l_(0)(G^("ad "))l_{0}\left(G^{\text {ad }}\right)l0(Gad ) can be shown to be equal to 0 by the second axiom in the definition of a Shimura datum.
As K K KKK varies, we have a tower of locally symmetric spaces ( X K ) K X K K (X_(K))_(K)\left(X_{K}\right)_{K}(XK)K, on which a spherical Hecke algebra T T T\mathbb{T}T for G G GGG acts by correspondences. The systems of Hecke eigenvalues occurring in the cohomology groups H ( c ) ( X K , C ) H ( c ) ∗ X K , C H_((c))^(**)(X_(K),C)H_{(c)}^{*}\left(X_{K}, \mathbb{C}\right)H(c)∗(XK,C) or, equivalently, the maximal ideals of T T T\mathbb{T}T in the support of these cohomology groups, can be related to automorphic representations of G ( A f ) G A f G(A_(f))G\left(\mathbb{A}_{f}\right)G(Af) by work of Franke and Matsushima [29]. The goal of this section is to state a conjecture on the cohomology of locally symmetric spaces with torsion coefficients F F ℓ F_(ℓ)\mathbb{F}_{\ell}Fℓ, where ℓ ℓ\ellℓ is a prime number. This conjecture is formulated in [25] (see the discussion around Conjecture 3.3) and in [12, CONJECTURE B]. Roughly, it says that the part of the cohomology outside the range of degrees [ q 0 , q 0 + l 0 ] q 0 , q 0 + l 0 [q_(0),q_(0)+l_(0)]\left[q_{0}, q_{0}+l_{0}\right][q0,q0+l0] is somehow degenerate. Note that this range of degrees is symmetric about the middle 1 2 dim R X 1 2 dim R ⁡ X (1)/(2)dim_(R)X\frac{1}{2} \operatorname{dim}_{\mathbb{R}} X12dimR⁡X of the total range of cohomology and, in the Shimura variety case, it equals the middle degree of cohomology.
To formulate this more precisely, we use the notion of a non-Eisenstein maximal ideal in the Hecke algebra, for which we need to pass to the Galois side of the global Langlands correspondence. For simplicity, we will restrict our formulation to the case of G = G = G=G=G= Res F / Q G L n Res F / Q ⁡ G L n Res_(F//Q)GL_(n)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{n}ResF/Q⁡GLn for some number field F F FFF, although the conjecture makes sense more generally. Let T T T\mathbb{T}T be the abstract spherical Hecke algebra away from a finite set S S SSS of primes of F F FFF and let m T m ⊂ T msubT\mathfrak{m} \subset \mathbb{T}m⊂T be a maximal ideal in the support of H ( c ) ( X K , F ) H ( c ) ∗ X K , F â„“ H_((c))^(**)(X_(K),F_(â„“))H_{(c)}^{*}\left(X_{K}, \mathbb{F}_{\ell}\right)H(c)∗(XK,Fâ„“). Assume that there exists a continuous, semisimple Galois representation ρ ¯ m : Gal ( F ¯ / F ) G L n ( F ¯ ) ρ ¯ m : Gal ⁡ ( F ¯ / F ) → G L n F ¯ â„“ bar(rho)_(m):Gal( bar(F)//F)rarrGL_(n)( bar(F)_(â„“))\bar{\rho}_{\mathfrak{m}}: \operatorname{Gal}(\bar{F} / F) \rightarrow \mathrm{GL}_{n}\left(\overline{\mathbb{F}}_{\ell}\right)ρ¯m:Gal⁡(F¯/F)→GLn(F¯ℓ) associated with m m m\mathfrak{m}m : by this, we mean that ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m is unramified at all the primes of F F FFF away from the finite set S S SSS, and that, at any prime away from S S SSS, the Satake parameters of m m m\mathfrak{m}m match the Frobenius eigenvalues of ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m. (The precise condition is in terms of the characteristic polynomial of ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m applied to the Frobenius at such a prime and depends on various choices of normalizations. See, for example, [1, THEOREM 2.3.5] for a precise formulation.) Since the Galois representation is assumed to be semisimple and we are working with Res F / Q G L n Res F / Q ⁡ G L n Res_(F//Q)GL_(n)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{n}ResF/Q⁡GLn, this property will characterize ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m by the Cebotarev density theorem and the Brauer-Nesbitt theorem. We say that m m m\mathfrak{m}m is non-Eisenstein if such a ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m is absolutely irreducible.
The existence of ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m as above should be thought of as a mod mod â„“ modâ„“\bmod \ellmodâ„“ version of the global Langlands correspondence, in the automorphic-to-Galois direction; in the case F = Q F = Q F=QF=\mathbb{Q}F=Q, this existence was conjectured by Ash [4]. The striking part of this conjecture is that it should apply to torsion classes in the cohomology of locally symmetric spaces, not just to those classes that lift to characteristic zero, and which can be related to automorphic representations of G G GGG. For general number fields, the existence of such Galois representations seems out of reach at the moment, even for classes in characteristic zero!
However, let F F FFF be a CM field: using nonstandard terminology, we mean that F F FFF is either a totally real field or a totally complex quadratic extension thereof. In this case, Scholze constructed such Galois representations in the breakthrough paper [53]. This strengthened previous work [33] that applied to cohomology with Q Q â„“ Q_(â„“)\mathbb{Q}_{\ell}Qâ„“-coefficients. Both these results relied, in turn, on the construction of Galois representations in the self-dual case, due to many people, including Clozel, Kottwitz, Harris-Taylor [34], Shin [61], and ChenevierHarris [21].
We can now state the promised vanishing conjecture for the cohomology of locally symmetric spaces with F F â„“ F_(â„“)\mathbb{F}_{\ell}Fâ„“-coefficients.
Conjecture 2.2. Assume that m T m ⊂ T msubT\mathfrak{m} \subset \mathbb{T}m⊂T is a non-Eisenstein maximal ideal in the support of H ( c ) ( X K , F ) H ( c ) ∗ X K , F ℓ H_((c))^(**)(X_(K),F_(ℓ))H_{(c)}^{*}\left(X_{K}, \mathbb{F}_{\ell}\right)H(c)∗(XK,Fℓ). Then H ( c ) i ( X K , F ) m 0 H ( c ) i X K , F ℓ m ≠ 0 H_((c))^(i)(X_(K),F_(ℓ))_(m)!=0H_{(c)}^{i}\left(X_{K}, \mathbb{F}_{\ell}\right)_{\mathfrak{m}} \neq 0H(c)i(XK,Fℓ)m≠0 only if i [ q 0 , q 0 + l 0 ] i ∈ q 0 , q 0 + l 0 i in[q_(0),q_(0)+l_(0)]i \in\left[q_{0}, q_{0}+l_{0}\right]i∈[q0,q0+l0].
In the two examples discussed in Example 2.1, this conjecture can be verified "by hand," since one only needs to control cohomology in degree 0 (the top degree of cohomology can be controlled using Poincaré duality). In the case of G L 2 / Q G L 2 / Q GL_(2)//Q\mathrm{GL}_{2} / \mathbb{Q}GL2/Q, one can show that the systems of Hecke eigenvalues m m m\mathfrak{m}m in the support of H 0 ( X K , F ) H 0 X K , F ℓ H^(0)(X_(K),F_(ℓ))H^{0}\left(X_{K}, \mathbb{F}_{\ell}\right)H0(XK,Fℓ) satisfy
(2.1) ρ ¯ m χ χ c y c l o χ (2.1) ρ ¯ m ≃ χ ⊕ χ c y c l o â‹… χ {:(2.1) bar(rho)_(m)≃chi o+chi_(cyclo)*chi:}\begin{equation*} \bar{\rho}_{\mathfrak{m}} \simeq \chi \oplus \chi_{\mathrm{cyclo}} \cdot \chi \tag{2.1} \end{equation*}(2.1)ρ¯m≃χ⊕χcyclo⋅χ
where χ χ chi\chiχ is a suitable mod mod â„“ modâ„“\bmod \ellmodâ„“ character of Gal ( Q ¯ / Q ) Gal ⁡ ( Q ¯ / Q ) Gal( bar(Q)//Q)\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})Gal⁡(Q¯/Q) and χ cyclo : Gal ( Q ¯ / Q ) F × Ï‡ cyclo  : Gal ⁡ ( Q ¯ / Q ) → F â„“ × chi_("cyclo "):Gal( bar(Q)//Q)rarrF_(â„“)^(xx)\chi_{\text {cyclo }}: \operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q}) \rightarrow \mathbb{F}_{\ell}^{\times}χcyclo :Gal⁡(Q¯/Q)→Fℓ×is the mod mod â„“ modâ„“\bmod \ellmodâ„“ cyclotomic character. Later, we will introduce a local genericity condition at an auxiliary prime p p ≠ â„“ p!=â„“p \neq \ellp≠ℓ and we will see that the ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m in (2.1) also fail to satisfy genericity everywhere. In addition to these and a few more low-dimensional examples, one can also consider the analogue of Conjecture 2.2 for H ( c ) ( X K , Q ) H ( c ) ∗ X K , Q â„“ H_((c))^(**)(X_(K),Q_(â„“))H_{(c)}^{*}\left(X_{K}, \mathbb{Q}_{\ell}\right)H(c)∗(XK,Qâ„“). This analogue is related to Arthur's conjectures on the cohomology of locally symmetric spaces [3] and can be verified for G L n G L n GL_(n)\mathrm{GL}_{n}GLn over C M C M CM\mathrm{CM}CM fields using work of Franke and Borel-Wallach (see [1, theOREM 2.4.9]).
Conjecture 2.2 is motivated by the Calegari-Geraghty enhancement [12] of the classical Taylor-Wiles method for proving automorphy lifting theorems. The classical method works well in settings where the (co)homology of locally symmetric spaces is concentrated in one degree, for example, for G L 2 / Q G L 2 / Q GL_(2)//Q\mathrm{GL}_{2} / \mathbb{Q}GL2/Q after localizing at a non-Eisenstein maximal ideal, or for definite unitary groups over totally real fields. In general, however, a certain numerical coincidence that is used to compare the Galois and automorphic sides breaks down. Calegari and Geraghty had a significant insight: they reinterpret the failure of the numerical coincidence in terms of the invariant l 0 l 0 l_(0)l_{0}l0. More precisely, l 0 l 0 l_(0)l_{0}l0 arises naturally from a computation on the Galois side, and the commutative algebra underlying the method can be adjusted if one knows that the cohomology on the automorphic side, after localizing at a non-Eisenstein maximal ideal, is concentrated in a range of degrees of length at most l 0 l 0 l_(0)l_{0}l0. For an overview of the key ideas involved in the Calegari-Geraghty method, see [10, §10].
In the case of Shimura varieties, Conjecture 2.2 predicts that the non-Eisenstein part of the cohomology with F F â„“ F_(â„“)\mathbb{F}_{\ell}Fâ„“-coefficients is concentrated in the middle degree. The initial progress on this conjecture in the Shimura variety setting had rather strong additional assumptions: for example, one needed â„“ â„“\ellâ„“ to be an unramified prime for the Shimura datum and K K â„“ K_(â„“)K_{\ell}Kâ„“ to be hyperspecial, as in the work of Dimitrov [23] and Lan-Suh [40,41]. The theory of perfectoid Shimura varieties and their associated Hodge-Tate period morphism has been a game-changer in this area. For the rest of this article, we will discuss more recent progress on Conjecture 2.2 and related questions in the special case of Shimura varieties, as well as applications that go beyond the setting of Shimura varieties.

3. THE HODGE-TATE PERIOD MORPHISM

The Hodge-Tate period morphism was introduced by Scholze in his breakthrough paper [53] and it was subsequently refined in [17]. It gives an entirely new way to think about the geometry and cohomology of Shimura varieties. In the past decade, it had numerous striking applications to the Langlands programme: to Scholze's construction of Galois representations for torsion classes, to the vanishing theorems discussed in Sections 4 and 5, to the construction of higher Coleman theory by Boxer and Pilloni [8], and to a radically new approach to the Fontaine-Mazur conjecture due to Pan [48].
For simplicity, let us consider a Shimura datum ( G , X ) ( G , X ) (G,X)(G, X)(G,X) of of Hodge type. By this, we mean that ( G , X ) ( G , X ) (G,X)(G, X)(G,X) admits a closed embedding into a Siegel datum ( G ~ , X ~ ) ( G ~ , X ~ ) ( tilde(G), tilde(X))(\tilde{G}, \tilde{X})(G~,X~), where G ~ = G S p 2 n G ~ = G S p 2 n tilde(G)=GSp_(2n)\tilde{G}=\mathrm{GSp}_{2 n}G~=GSp2n, for some n Z 1 n ∈ Z ≥ 1 n inZ_( >= 1)n \in \mathbb{Z}_{\geq 1}n∈Z≥1, and X ~ X ~ tilde(X)\tilde{X}X~ is as in (1.1). For example, ( G , X ) ( G , X ) (G,X)(G, X)(G,X) could be a Shimura datum of P E L P E L PELP E LPEL type arising from a unitary similitude group: the corresponding Shimura varieties will represent a moduli problem of abelian varieties equipped with extra structures (polarizations, endomorphisms, and level structures). This unitary case will be the main example to keep in mind, as this will also play a central role in Section 4.
For some representative h X h ∈ X h in Xh \in Xh∈X, we consider the Hodge cocharacter
μ = h × R C | 1 st G m factor : G m , C G C μ = h × R C 1  st  G m  factor  : G m , C → G C mu=h xx_(R)C|_(1" st "G_(m)" factor "):G_(m,C)rarrG_(C)\mu=h \times\left._{\mathbb{R}} \mathbb{C}\right|_{1 \text { st } \mathbb{G}_{m} \text { factor }}: \mathbb{G}_{m, \mathbb{C}} \rightarrow G_{\mathbb{C}}μ=h×RC|1 st Gm factor :Gm,C→GC
The axioms in the definition of the Shimura datum imply that μ μ mu\muμ is minuscule. The reflex field E E EEE is the field of definition of the conjugacy class { μ } { μ } {mu}\{\mu\}{μ}; it is a finite extension of Q Q Q\mathbb{Q}Q and the corresponding Shimura varieties admit canonical models over E E EEE. The cocharacter μ μ mu\muμ also determines two opposite parabolic subgroups P μ std P μ std  P_(mu)^("std ")P_{\mu}^{\text {std }}Pμstd  and P μ P μ P_(mu)P_{\mu}Pμ, whose conjugacy classes are defined over E E EEE. These are given by
P μ std = { g G lim t ad ( μ ( t ) ) g exists } , P μ = { g G lim t 0 ad ( μ ( t ) ) g exists } P μ std  = g ∈ G ∣ lim t → ∞   ad ⁡ ( μ ( t ) ) g  exists  , P μ = g ∈ G ∣ lim t → 0   ad ⁡ ( μ ( t ) ) g  exists  P_(mu)^("std ")={g in G∣lim_(t rarr oo)ad(mu(t))g" exists "},quadP^(mu)={g in G∣lim_(t rarr0)ad(mu(t))g" exists "}P_{\mu}^{\text {std }}=\left\{g \in G \mid \lim _{t \rightarrow \infty} \operatorname{ad}(\mu(t)) g \text { exists }\right\}, \quad P^{\mu}=\left\{g \in G \mid \lim _{t \rightarrow 0} \operatorname{ad}(\mu(t)) g \text { exists }\right\}Pμstd ={g∈G∣limt→∞ad⁡(μ(t))g exists },Pμ={g∈G∣limt→0ad⁡(μ(t))g exists }
We let F l std F l std  Fl^("std ")\mathrm{Fl}^{\text {std }}Flstd  and F l F l Fl\mathrm{Fl}Fl denote the associated flag varieties, which are also defined over E E EEE.
Here is a more moduli-theoretic way to think about these the two parabolics. The chosen symplectic embedding ( G , X ) ( G ~ , X ~ ) ( G , X ) ↪ ( G ~ , X ~ ) (G,X)↪( tilde(G), tilde(X))(G, X) \hookrightarrow(\tilde{G}, \tilde{X})(G,X)↪(G~,X~) gives rise to a faithful representation V V VVV of G G GGG. The embedding also gives rise to an abelian scheme A K A K A_(K)A_{K}AK over the Shimura variety S K S K S_(K)S_{K}SK at some level K = G ( A f ) K ~ K = G A f ∩ K ~ K=G(A_(f))nn tilde(K)K=G\left(\mathbb{A}_{f}\right) \cap \tilde{K}K=G(Af)∩K~, obtained by restricting the universal abelian scheme over the Siegel modular variety at level K ~ G ~ ( A f ) K ~ ⊂ G ~ A f tilde(K)sub tilde(G)(A_(f))\tilde{K} \subset \tilde{G}\left(\mathbb{A}_{f}\right)K~⊂G~(Af). The cocharacter μ μ mu\muμ induces a grading of V C V C V_(C)V_{\mathbb{C}}VC,
which in turn defines two filtrations on V C V C V_(C)V_{\mathbb{C}}VC, a descending one Fil ∙ ^(∙){ }^{\bullet}∙ and an ascending one Fil â‹… *\cdotâ‹… The parabolic P μ std P μ std  P_(mu)^("std ")P_{\mu}^{\text {std }}Pμstd  is the stabilizer of Fil ∙ ^(∙){ }^{\bullet}∙, which is morally the Hodge-de Rham filtration on the Betti cohomology of A K A K A_(K)A_{K}AK. There is a holomorphic, G ( R ) G ( R ) G(R)G(\mathbb{R})G(R)-equivariant embedding
(3.1) π d R : X F l s t d ( C ) = G ( C ) / P μ s t d (3.1) Ï€ d R : X ↪ F l s t d ( C ) = G ( C ) / P μ s t d {:(3.1)pi_(dR):X↪Fl^(std)(C)=G(C)//P_(mu)^(std):}\begin{equation*} \pi_{\mathrm{dR}}: X \hookrightarrow \mathrm{Fl}^{\mathrm{std}}(\mathbb{C})=G(\mathbb{C}) / P_{\mu}^{\mathrm{std}} \tag{3.1} \end{equation*}(3.1)Ï€dR:X↪Flstd(C)=G(C)/Pμstd
called the Borel embedding, defined by h F i l ( μ h ) h ↦ F i l ∙ μ h h|->Fil^(∙)(mu_(h))h \mapsto \mathrm{Fil}^{\bullet}\left(\mu_{h}\right)h↦Fil∙(μh). The axioms of a Shimura datum imply that X X XXX is a variation of polarisable Hodge structures of abelian varieties. Modulitheoretically, π d R Ï€ d R pi_(dR)\pi_{\mathrm{dR}}Ï€dR sends a Hodge structure, such as
H 1 ( A , Q ) Q C H 0 ( A , Ω A 1 ) H 1 ( A , O A ) H 1 ( A , Q ) ⊗ Q C ≃ H 0 A , Ω A 1 ⊕ H 1 A , O A H^(1)(A,Q)ox_(Q)C≃H^(0)(A,Omega_(A)^(1))o+H^(1)(A,O_(A))H^{1}(A, \mathbb{Q}) \otimes_{\mathbb{Q}} \mathbb{C} \simeq H^{0}\left(A, \Omega_{A}^{1}\right) \oplus H^{1}\left(A, \mathcal{O}_{A}\right)H1(A,Q)⊗QC≃H0(A,ΩA1)⊕H1(A,OA)
to the associated Hodge-de Rham filtration, e.g., H 0 ( A , Ω A 1 ) H 1 ( A , Q ) Q C H 0 A , Ω A 1 ⊂ H 1 ( A , Q ) ⊗ Q C H^(0)(A,Omega_(A)^(1))subH^(1)(A,Q)oxQCH^{0}\left(A, \Omega_{A}^{1}\right) \subset H^{1}(A, \mathbb{Q}) \otimes \mathbb{Q} \mathbb{C}H0(A,ΩA1)⊂H1(A,Q)⊗QC. The embedding π d R Ï€ d R pi_(dR)\pi_{\mathrm{dR}}Ï€dR is an example of a period morphism. Historically, it has played an important role in the construction of canonical models of automorphic vector bundles over E E EEE (or even integrally), such as in work of Harris and Milne.
On the other hand, the parabolic subgroup P μ P μ P_(mu)P_{\mu}Pμ is the stabilizer of the ascending filtration Fil. This gives rise to an antiholomorphic embedding
X Fl ( C ) = G ( C ) / P μ X ↪ Fl ⁡ ( C ) = G ( C ) / P μ X↪Fl(C)=G(C)//P_(mu)X \hookrightarrow \operatorname{Fl}(\mathbb{C})=G(\mathbb{C}) / P_{\mu}X↪Fl⁡(C)=G(C)/Pμ
Morally, P μ P μ P_(mu)P_{\mu}Pμ is the stabilizer of the Hodge-Tate filtration on the p p ppp-adic étale cohomology of A K A K A_(K)A_{K}AK. The Hodge-Tate period morphism will be a p p ppp-adic analogue of the embedding (3.1) (or perhaps of the embedding (3.2), depending on one's perspective).
Let p p ppp be a rational prime, p p p ∣ p p∣p\mathfrak{p} \mid pp∣p a prime of E E EEE, and let C C CCC be the completion of an algebraic closure of E p E p E_(p)E_{\mathfrak{p}}Ep. We consider the adic spaces K ∮ K oint_(K)\oint_{K}∮K and F F ℓ Fℓ\mathscr{F} \ellFℓ over Spa ( C , O C ) Spa ⁡ C , O C Spa(C,O_(C))\operatorname{Spa}\left(C, \mathcal{O}_{C}\right)Spa⁡(C,OC) corresponding to the algebraic varieties S K S K S_(K)S_{K}SK and F l F l Fl\mathrm{Fl}Fl over E E EEE. A striking result of Scholze shows that the tower of Shimura varieties ( S K p K p ) K p S K p K p K p (S_(K^(p)K_(p)))_(K_(p))\left(S_{K^{p} K_{p}}\right)_{K_{p}}(SKpKp)Kp acquires the structure of a perfectoid space (in the sense of [51]) as K p K p K_(p)K_{p}Kp varies over compact open subgroups of G ( Q p ) G Q p G(Q_(p))G\left(\mathbb{Q}_{p}\right)G(Qp). More precisely, the following result was established in [ 53 , $ 3 , 4 ] [ 53 , $ 3 , 4 ] [53,$3,4][53, \$ 3,4][53,$3,4] and later refined in [ 17 , $ 2 ] [ 17 , $ 2 ] [17,$2][17, \$ 2][17,$2], by correctly identifying the target of the Hodge-Tate period morphism.
Theorem 3.1. There exists a unique perfectoid space S K p S K p S_(K^(p))S_{K^{p}}SKp satisfying S K p lim K p S K p K p , 2 S K p ∼ lim K p   S K p K p , 2 S_(K^(p))∼_(lim_(K_(p)))S_(K^(p)K_(p)),^(2)S_{K^{p}} \sim \underset{\lim _{K_{p}}}{ } \mathcal{S}_{K^{p} K_{p}},{ }^{2}SKp∼limKpSKpKp,2 in the sense of [55, DEFINITION 2.4.1], and a G ( Q p ) G Q p G(Q_(p))G\left(\mathbb{Q}_{p}\right)G(Qp)-equivariant morphism of adic spaces
π H T : S K p F Ï€ H T : S K p → F â„“ pi_(HT):S_(K^(p))rarrFâ„“\pi_{\mathrm{HT}}: S_{K^{p}} \rightarrow \mathscr{F} \ellÏ€HT:SKp→Fâ„“
Moreover, π H T Ï€ H T pi_(HT)\pi_{\mathrm{HT}}Ï€HT is equivariant for the usual action of Hecke operators away from p p ppp on S K p S K p S_(K^(p))S_{K^{p}}SKp and their trivial action on F F â„“ Fâ„“\mathscr{F} \ellFâ„“.
In the Siegel case G = G S p 2 n / Q G = G S p 2 n / Q G=GSp_(2n)//QG=\mathrm{GSp}_{2 n} / \mathbb{Q}G=GSp2n/Q, one can describe the Hodge-Tate period morphism π H T Ï€ H T pi_(HT)\pi_{\mathrm{HT}}Ï€HT from a moduli-theoretic perspective as follows. An abelian variety A / C A / C A//CA / CA/C, equipped with a trivialization T p A Z p 2 n T p A ≃ Z p 2 n T_(p)A≃Z_(p)^(2n)T_{p} A \simeq \mathbb{Z}_{p}^{2 n}TpA≃Zp2n will be sent to the first piece of the Hodge-Tate filtration
Lie A T p A Z p C C 2 n A ⊂ T p A ⊗ Z p C ≃ C 2 n A subT_(p)Aox_(Z_(p))C≃C^(2n)A \subset T_{p} A \otimes_{\mathbb{Z}_{p}} C \simeq C^{2 n}A⊂TpA⊗ZpC≃C2n.
2 It is enough to consider the Shimura varieties as adic spaces over E p E p E_(p)E_{\mathfrak{p}}Ep and the tower still acquires a perfectoid structure in a noncanonical way. We work over C C CCC for simplicity and also because this gives rise to the étale cohomology groups we want to understand.
Dually, one has the Hodge-Tate filtration on the p p ppp-adic étale cohomology of A A AAA :
(3.3) 0 H 1 ( A , O A ) H e t 1 ( A , Z p ) Z p C H 0 ( A , Ω A / C 1 ) ( 1 ) 0 (3.3) 0 → H 1 A , O A → H e t 1 A , Z p ⊗ Z p C → H 0 A , Ω A / C 1 ( − 1 ) → 0 {:(3.3)0rarrH^(1)(A,O_(A))rarrH_(et)^(1)(A,Z_(p))ox_(Z_(p))C rarrH^(0)(A,Omega_(A//C)^(1))(-1)rarr0:}\begin{equation*} 0 \rightarrow H^{1}\left(A, \mathcal{O}_{A}\right) \rightarrow H_{\mathrm{et}}^{1}\left(A, \mathbb{Z}_{p}\right) \otimes_{\mathbb{Z}_{p}} C \rightarrow H^{0}\left(A, \Omega_{A / C}^{1}\right)(-1) \rightarrow 0 \tag{3.3} \end{equation*}(3.3)0→H1(A,OA)→Het1(A,Zp)⊗ZpC→H0(A,ΩA/C1)(−1)→0
where ( 1 ) ( − 1 ) (-1)(-1)(−1) denotes a Tate twist (which is important for keeping track of the Galois action). To show that the morphism defined this way on Spa ( C , C + ) Spa ⁡ C , C + Spa(C,C^(+))\operatorname{Spa}\left(C, C^{+}\right)Spa⁡(C,C+)-points comes from a morphism of adic spaces, it is important to know that the filtration (3.3) varies continuously. At the same time, to extend the result to Shimura varieties of Hodge type and to cut down the image to F F â„“ Fâ„“\mathscr{F} \ellFâ„“, one needs to keep track of Hodge tensors carefully. Both problems are solved via relative p p ppp-adic Hodge theory for the morphism A K S K A K → S K A_(K)rarrS_(K)\mathscr{A}_{K} \rightarrow S_{K}AK→SK, where A K A K A_(K)\mathcal{A}_{K}AK is the restriction to ς K Ï‚ K Ï‚_(K)\varsigma_{K}Ï‚K of a universal abelian scheme over an ambient Siegel modular variety. See [13, §3] for an overview.
Theorem 3.1 can be extended to minimal and toroidal compactifications of Siegel modular varieties, cf. [53] and [49]. Moreover, there is a natural affinoid cover of F F â„“ Fâ„“\mathscr{F} \ellFâ„“ such that the preimage under π H T Ï€ H T pi_(HT)\pi_{\mathrm{HT}}Ï€HT of each affinoid in the cover is an affinoid perfectoid subspace of S K p S K p ∗ S_(K^(p))^(**)S_{K^{p}}^{*}SKp∗. The underlying reason for this is the fact that the partial minimal compactification of the ordinary locus is affine. The perfectoid structure on S K p S K p ∗ S_(K^(p))^(**)S_{K^{p}}^{*}SKp∗ and the affinoid nature of the Hodge-Tate period morphism play an important role in Scholze's p p ppp-adic interpolation argument, that is key for the construction of Galois representations associated with torsion classes. See also [44] for an exposition of the main ideas.
Theorem 3.1 can also be extended to minimal and toroidal compactifications of Shimura varieties of Hodge type and even abelian type, cf. [32, 58] and [8], although there are some technical issues at the boundary. For example, the cleanest formulation currently available in full generality is that the relationship S K p = lim K p S K p K p S K p ∗ = lim ⟵ K p   S K p K p ∗ S_(K^(p))^(**)=lim_(longleftarrowK_(p))S_(K^(p)K_(p))^(**)S_{K^{p}}^{*}=\lim _{\longleftarrow K_{p}} S_{K^{p} K_{p}}^{*}SKp∗=lim⟵KpSKpKp∗, for a perfectoid space S K p S K p ∗ S_(K^(p))^(**)\mathcal{S}_{K^{p}}^{*}SKp∗, holds in Scholze's category of diamonds [54].
Example 3.2. To see where the perfectoid structure on S K p S K p S_(K^(p))S_{K^{p}}SKp comes from, it is instructive to consider the case of modular curves and study the geometry of their special fibers: we are particularly interested in the geometry of the so-called Deligne-Rapoport model. Set G = G = G=G=G= G L 2 / Q G L 2 / Q GL_(2)//Q\mathrm{GL}_{2} / \mathbb{Q}GL2/Q. Let K p 0 = G L 2 ( Z p ) K p 0 = G L 2 Z p K_(p)^(0)=GL_(2)(Z_(p))K_{p}^{0}=\mathrm{GL}_{2}\left(\mathbb{Z}_{p}\right)Kp0=GL2(Zp), the hyperspecial compact open subgroup and let S ¯ K p K p 0 / F p S ¯ K p K p 0 / F p bar(S)_(K^(p)K_(p)^(0))//F_(p)\bar{S}_{K^{p} K_{p}^{0}} / \mathbb{F}_{p}S¯KpKp0/Fp be the special fiber of the integral model over Z ( p ) Z ( p ) Z_((p))\mathbb{Z}_{(p)}Z(p) of the modular curve at this level. This is a smooth curve over F p F p F_(p)\mathbb{F}_{p}Fp that represents a moduli problem ( E , α ) ( E , α ) (E,alpha)(E, \alpha)(E,α) of elliptic curves equipped with prime-to- p p ppp level structures (determined by the prime-to- p p ppp level K p K p K^(p)K^{p}Kp ). The isogeny class of the p p ppp-divisible group E [ p ] E p ∞ E[p^(oo)]E\left[p^{\infty}\right]E[p∞] induces the Newton stratification
(3.4) S ¯ K p K p 0 = S ¯ K p K p 0 o r d S ¯ K p K p 0 s s (3.4) S ¯ K p K p 0 = S ¯ K p K p 0 o r d ⊔ S ¯ K p K p 0 s s {:(3.4) bar(S)_(K^(p)K_(p)^(0))= bar(S)_(K^(p)K_(p)^(0))^(ord)⊔ bar(S)_(K^(p)K_(p)^(0))^(ss):}\begin{equation*} \bar{S}_{K^{p} K_{p}^{0}}=\bar{S}_{K^{p} K_{p}^{0}}^{\mathrm{ord}} \sqcup \bar{S}_{K^{p} K_{p}^{0}}^{\mathrm{ss}} \tag{3.4} \end{equation*}(3.4)S¯KpKp0=S¯KpKp0ord⊔S¯KpKp0ss
into an open dense ordinary stratum S ¯ K p K p 0 ord S ¯ K p K p 0 ord  bar(S)_(K^(p)K_(p)^(0))^("ord ")\bar{S}_{K^{p} K_{p}^{0}}^{\text {ord }}S¯KpKp0ord  (where E [ p ] E p ∞ E[p^(oo)]E\left[p^{\infty}\right]E[p∞] is isogenous to μ p × Q p / Z p μ p ∞ × Q p / Z p mu_(p^(oo))xxQ_(p)//Z_(p)\mu_{p^{\infty}} \times \mathbb{Q}_{p} / \mathbb{Z}_{p}μp∞×Qp/Zp ) and a closed supersingular stratum S ¯ K p s s K p 0 S ¯ K p s s K p 0 bar(S)_(K^(p))^(ss)K_(p)^(0)\bar{S}_{K^{p}}^{\mathrm{ss}} K_{p}^{0}S¯KpssKp0 consisting of finitely many points (where E [ p ] E p ∞ E[p^(oo)]E\left[p^{\infty}\right]E[p∞] is connected).
Now let K p 1 G L 2 ( Q p ) K p 1 ⊂ G L 2 Q p K_(p)^(1)subGL_(2)(Q_(p))K_{p}^{1} \subset \mathrm{GL}_{2}\left(\mathbb{Q}_{p}\right)Kp1⊂GL2(Qp) be the Iwahori subgroup and S ¯ K p K p 1 / F p S ¯ K p K p 1 / F p bar(S)_(K^(p)K_(p)^(1))//F_(p)\bar{S}_{K^{p} K_{p}^{1}} / \mathbb{F}_{p}S¯KpKp1/Fp be the special fiber of the integral model of the modular curve at this level. This represents a moduli problem ( E , α , D ) ( E , α , D ) (E,alpha,D)(E, \alpha, D)(E,α,D) of elliptic curves equipped with prime-to- p p ppp level structures and also with a level structure at p p ppp given by a finite flat subgroup scheme D E [ p ] D ⊂ E [ p ] D sub E[p]D \subset E[p]D⊂E[p] of order p p ppp. Again, we have the
preimage of the Newton stratification S ¯ K p K p 1 = S ¯ K p K p 1 ord S ¯ K p K p 1 s s S ¯ K p K p 1 = S ¯ K p K p 1 ord  ⊔ S ¯ K p K p 1 s s bar(S)_(K^(p)K_(p)^(1))= bar(S)_(K^(p)K_(p)^(1))^("ord ")⊔ bar(S)_(K^(p)K_(p)^(1))^(ss)\bar{S}_{K^{p} K_{p}^{1}}=\bar{S}_{K^{p} K_{p}^{1}}^{\text {ord }} \sqcup \bar{S}_{K^{p} K_{p}^{1}}^{\mathrm{ss}}S¯KpKp1=S¯KpKp1ord ⊔S¯KpKp1ss. The modular curve at this level is not smooth, but rather a union of irreducible components that intersect transversely at the finitely many supersingular points.
The open and dense ordinary locus S ¯ K p K p 1 ord S ¯ K p K p 1 ord  bar(S)_(K^(p)K_(p)^(1))^("ord ")\bar{S}_{K^{p} K_{p}^{1}}^{\text {ord }}S¯KpKp1ord  is a disjoint union of two KottwitzRapoport strata: the one where D μ p D ≃ μ p D≃mu_(p)D \simeq \mu_{p}D≃μp and the one where D Z / p Z D ≃ Z / p Z D≃Z//pZD \simeq \mathbb{Z} / p \mathbb{Z}D≃Z/pZ. Both of these Kottwitz-Rapoport strata can be shown to be abstractly isomorphic to the ordinary stratum at hyperspecial level. If we restrict the natural forgetful map S ¯ K p K p 1 ord S ¯ K p K p 0 ord S ¯ K p K p 1 ord  → S ¯ K p K p 0 ord  bar(S)_(K^(p)K_(p)^(1))^("ord ")rarr bar(S)_(K^(p)K_(p)^(0))^("ord ")\bar{S}_{K^{p} K_{p}^{1}}^{\text {ord }} \rightarrow \bar{S}_{K^{p} K_{p}^{0}}^{\text {ord }}S¯KpKp1ord →S¯KpKp0ord  to the Kottwitz-Rapoport stratum where D Z / p Z D ≃ Z / p Z D≃Z//pZD \simeq \mathbb{Z} / p \mathbb{Z}D≃Z/pZ, the map can be identified (up to an isomorphism) with the geometric Frobenius. (The restriction of the map to the Kottwitz-Rapoport stratum where D μ p D ≃ μ p D≃mu_(p)D \simeq \mu_{p}D≃μp is an isomorphism.)
On the adic generic fiber, one can extend this picture to an anticanonical ordinary tower, where the transition morphisms reduce modulo p p ppp to (powers of) the geometric Frobenius, giving a perfectoid space in the limit. To extend beyond the ordinary locus, Scholze uses the theory of the canonical subgroup, the action of G L 2 ( Q p ) G L 2 Q p GL_(2)(Q_(p))\mathrm{GL}_{2}\left(\mathbb{Q}_{p}\right)GL2(Qp) at infinite level, and a rudimentary form of the Hodge-Tate period morphism that is just defined on the underlying topological spaces.
The above strategy generalizes relatively cleanly to higher-dimensional Siegel modular varieties, modulo subtleties at the boundary. To extend Theorem 3.1 to general Shimura varieties of Hodge type, Scholze considers an embedding at infinite level into a Siegel modular variety. It is surprisingly subtle to understand directly the perfectoid structure on a general Shimura variety of Hodge type (especially in the case when G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp is nonsplit) and this is related to the discussion in Section 5. This is also related to the fact that the geometry of the EKOR stratification is more intricate when G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp is nonsplit.
For simplicity, let us now assume that ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is a Shimura datum of PEL type and that p p ppp is an unramified prime for this Shimura datum. Recall the Kottwitz set B ( G ) B ( G ) B(G)B(G)B(G) classifying isocrystals with G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure. The Hodge cocharacter μ μ mu\muμ defines a subset B ( G , μ 1 ) B G , μ − 1 ⊂ B(G,mu^(-1))subB\left(G, \mu^{-1}\right) \subsetB(G,μ−1)⊂ B ( G ) B ( G ) B(G)B(G)B(G) of μ 1 μ − 1 mu^(-1)\mu^{-1}μ−1-admissible elements. The special fiber of the Shimura variety with hyperspecial level at p p ppp admits a Newton stratification
S ¯ K p K p 0 = b B ( G , μ 1 ) S ¯ K p K p 0 b S ¯ K p K p 0 = ⨆ b ∈ B G , μ − 1   S ¯ K p K p 0 b bar(S)_(K^(p)K_(p)^(0))=⨆_(b in B(G,mu^(-1))) bar(S)_(K^(p)K_(p)^(0))^(b)\bar{S}_{K^{p} K_{p}^{0}}=\bigsqcup_{b \in B\left(G, \mu^{-1}\right)} \bar{S}_{K^{p} K_{p}^{0}}^{b}S¯KpKp0=⨆b∈B(G,μ−1)S¯KpKp0b
into locally closed strata indexed by this subset. This stratification is in terms of isogeny classes of p p ppp-divisible groups with G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure and generalizes the stratification (3.4) from the modular curve case.
For each b B ( G , μ 1 ) b ∈ B G , μ − 1 b in B(G,mu^(-1))b \in B\left(G, \mu^{-1}\right)b∈B(G,μ−1), one can choose a (completely slope divisible) p p ppp-divisible group with G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure X b / F ¯ p X b / F ¯ p X_(b)// bar(F)_(p)\mathbb{X}_{b} / \overline{\mathbb{F}}_{p}Xb/F¯p and define the corresponding Oort central leaf. This is a
smooth closed subscheme C X b C X b C^(Xb)\mathscr{C}^{\mathbb{X} b}CXb of the Newton stratum S ¯ K p K p 0 b S ¯ K p K p 0 b bar(S)_(K^(p)K_(p)^(0))^(b)\bar{S}_{K^{p} K_{p}^{0}}^{b}S¯KpKp0b, such that the isomorphism class of the p p ppp-divisible group with G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure over each geometric point of the leaf is constant and equal to that of X b X b X_(b)\mathbb{X}_{b}Xb :
C X b = { x S ¯ K p K p 0 b A ¯ K p K p 0 [ p ] × κ ( x ¯ ) X b × κ ( x ¯ ) } C X b = x ∈ S ¯ K p K p 0 b ∣ A ¯ K p K p 0 p ∞ × κ ( x ¯ ) ≃ X b × κ ( x ¯ ) C^(X_(b))={x in bar(S)_(K^(p)K_(p)^(0))^(b)∣ bar(A)_(K^(p)K_(p)^(0))[p^(oo)]xx kappa(( bar(x)))≃X_(b)xx kappa(( bar(x)))}\mathscr{C}^{\mathbb{X}_{b}}=\left\{x \in \bar{S}_{K^{p} K_{p}^{0}}^{b} \mid \bar{A}_{K^{p} K_{p}^{0}}\left[p^{\infty}\right] \times \kappa(\bar{x}) \simeq \mathbb{X}_{b} \times \kappa(\bar{x})\right\}CXb={x∈S¯KpKp0b∣A¯KpKp0[p∞]×κ(x¯)≃Xb×κ(x¯)}
In general, there can be infinitely many leaves inside a given Newton stratum. Over each central leaf, one has the perfect Igusa variety Ig b / F ¯ p Ig b ⁡ / F ¯ p Ig^(b)// bar(F)_(p)\operatorname{Ig}^{b} / \bar{F}_{p}Igb⁡/F¯p, a profinite cover of C X b C X b C^(X^(b))\mathscr{C}^{\mathbb{X}^{b}}CXb which parametrizes trivializations of the universal p p ppp-divisible group with G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure.
Variants of Igusa varieties were introduced in [34] in the special case of Shimura varieties of Harris-Taylor type. They were defined more generally for Shimura varieties of PEL type by Mantovan [43] and their ℓ ℓ\ellℓ-adic cohomology was computed in many cases by Shin using a counting point formula [59-61]. All these authors consider Igusa varieties as profinite étale covers of central leaves, which trivialize the graded pieces of the slope filtration on the universal p p ppp-divisible group. Taking perfection gives a more elegant moduli-theoretic interpretation, while preserving ℓ ℓ\ellℓ-adic cohomology. However, the coherent cohomology of Igusa varieties is also important for defining and studying p p ppp-adic families of automorphic forms on G G GGG, as pioneered by Katz and Hida. Taking perfection is too crude for this purpose.
While the central leaf C X b C X b C^(X_(b))\mathscr{C}^{\mathbb{X}_{b}}CXb depends on the choice of X b X b X_(b)\mathbb{X}_{b}Xb in its isogeny class, one can show that the perfect Igusa variety I g b I g b Ig^(b)\mathrm{Ig}^{b}Igb only depends on the isogeny class: this follows from the equivalent moduli-theoretic description in [17, LEMMA 4.3.4] (see also [19, LEMMA 4.2.2], which keeps track of the extra structures more carefully). In particular, the pair ( G , μ ) ( G , μ ) (G,mu)(G, \mu)(G,μ) is not determined by the Igusa variety I g b I g b Ig^(b)\mathrm{Ig}^{b}Igb - it can happen that Igusa varieties that are a priori obtained from different Shimura varieties are isomorphic. See [19, THEOREM 4.2.4] for an example and [57] for a systematic analysis of this phenomenon in the function field setting.
Because I g b / F ¯ p I g b / F ¯ p Ig^(b)// bar(F)_(p)\mathrm{Ig}^{b} / \overline{\mathbb{F}}_{p}Igb/F¯p is perfect, the base change I g b × F ¯ p O C / p I g b × F ¯ p O C / p Ig^(b)xx bar(F)_(p)O_(C)//p\mathrm{Ig}^{b} \times \overline{\mathbb{F}}_{p} \mathcal{O}_{C} / pIgb×F¯pOC/p admits a canonical lift to a flat formal scheme over Spf O C Spf ⁡ O C Spf O_(C)\operatorname{Spf} \mathcal{O}_{C}Spf⁡OC. We let g b â„‘ g b â„‘g^(b)\mathfrak{\Im} \mathfrak{g}^{b}â„‘gb denote the adic generic fiber of this lift, which is a perfectoid space over Spa ( C , O C ) Spa ⁡ C , O C Spa(C,O_(C))\operatorname{Spa}\left(C, \mathcal{O}_{C}\right)Spa⁡(C,OC). The spaces I g b I g b Ig^(b)\mathrm{Ig}^{b}Igb and g b â„‘ g b â„‘g^(b)\mathfrak{\Im} \mathfrak{g}^{b}â„‘gb have naturally isomorphic â„“ â„“\ellâ„“-adic cohomology groups and they both have an action of a locally profinite group G b ( Q p ) G b Q p G_(b)(Q_(p))G_{b}\left(\mathbb{Q}_{p}\right)Gb(Qp), where G b G b G_(b)G_{b}Gb is an inner form of a Levi subgroup of G G GGG.
For each b B ( G , μ 1 ) b ∈ B G , μ − 1 b in B(G,mu^(-1))b \in B\left(G, \mu^{-1}\right)b∈B(G,μ−1), one can also consider the associated Rapoport-Zink space, a moduli space of p p ppp-divisible groups with G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure that is a local analogue of a Shimura variety. Concretely in the PEL case, one considers a moduli problem of p p ppp-divisible groups equipped with G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure, satisfying the Kottwitz determinant condition with respect to μ μ mu\muμ, and with a modulo p p ppp quasiisogeny to the fixed p p ppp-divisible group X b X b X_(b)\mathbb{X}_{b}Xb. This moduli problem was shown by Rapoport-Zink [50] to be representable by a formal scheme over Spf O E ˘ p Spf ⁡ O E ˘ p Spf O_(E^(˘)_(p))\operatorname{Spf} \mathcal{O}_{\breve{E}_{\mathfrak{p}}}Spf⁡OE˘p, where E ˘ p E ˘ p E^(˘)_(p)\breve{E}_{\mathfrak{p}}E˘p is the completion of the maximal unramified extension of E p E p E_(p)E_{\mathfrak{p}}Ep. We let M b M b M^(b)\mathcal{M}^{b}Mb denote the adic generic fiber of this formal scheme, 3 3 ^(3){ }^{3}3 base changed to Spa ( C , O C ) Spa ⁡ C , O C Spa(C,O_(C))\operatorname{Spa}\left(C, \mathcal{O}_{C}\right)Spa⁡(C,OC), and let M b M ∞ b M_(oo)^(b)\mathcal{M}_{\infty}^{b}M∞b denote the corresponding infinite-level Rapoport-Zink space. The latter object can be shown to be a perfectoid space using the techniques of [55], by which the infinite-level Rapoport-Zink space admits a local analogue of the Hodge-Tate period morphism
π H T b : M b F Ï€ H T b : M ∞ b → F â„“ pi_(HT)^(b):M_(oo)^(b)rarrFâ„“\pi_{\mathrm{HT}}^{b}: \mathcal{M}_{\infty}^{b} \rightarrow \mathscr{F} \ellÏ€HTb:M∞b→Fâ„“
It turns out that the geometry of π H T Ï€ H T pi_(HT)\pi_{\mathrm{HT}}Ï€HT is intricately tied up with the geometry of its local analogues π H T b Ï€ H T b pi_(HT)^(b)\pi_{\mathrm{HT}}^{b}Ï€HTb. The following result is a conceptually cleaner, infinite-level version of the Mantovan product formula established in [43], which describes Newton strata inside Shimura varieties in terms of a product of Igusa varieties and Rapoport-Zink spaces.
Theorem 3.3. There exists a Newton stratification
F = b B ( G , μ 1 ) F b F â„“ = ⨆ b ∈ B G , μ − 1   F â„“ b Fâ„“=⨆_(b in B(G,mu^(-1)))Fâ„“^(b)\mathscr{F} \ell=\bigsqcup_{b \in B\left(G, \mu^{-1}\right)} \mathscr{F} \ell^{b}Fâ„“=⨆b∈B(G,μ−1)Fâ„“b
into locally closed strata.
For each b B ( G , μ 1 ) b ∈ B G , μ − 1 b in B(G,mu^(-1))b \in B\left(G, \mu^{-1}\right)b∈B(G,μ−1), one can consider the Newton stratum K p b ∮ K p ∘ b oint_(K^(p))^(@b)\oint_{K^{p}}^{\circ b}∮Kp∘b as a locally closed subspace of the good reduction locus S K p S K p ∘ S_(K^(p))^(@)S_{K^{p}}^{\circ}SKp∘. There exists a Cartesian diagram of diamonds over Spd ( C , O C ) Spd ⁡ C , O C Spd(C,O_(C))\operatorname{Spd}\left(C, \mathcal{O}_{C}\right)Spd⁡(C,OC)
Moreover, each vertical map is a pro-étale torsor for the group diamond G ~ b G ~ b tilde(G)_(b)\tilde{G}_{b}G~b (identified with Aut G ( X ~ b ) Aut G ⁡ X ~ b Aut_(G)( widetilde(X)_(b))\operatorname{Aut}_{G}\left(\widetilde{\mathbb{X}}_{b}\right)AutG⁡(X~b), in the notation of [ 17 , $ 4 ] ) [ 17 , $ 4 ] {:[17,$4])\left.[17, \$ 4]\right)[17,$4]).
The decomposition into Newton strata is defined in [17, §3]. Morally, one first constructs a map of v-stacks F F â„“ → Fâ„“rarr\mathscr{F} \ell \rightarrowFℓ→ Bun G G _(G)_{G}G, where the latter is the v v vvv-stack of G G GGG-bundles on the Fargues-Fontaine curve. To construct this map of v-stacks, it is convenient to notice that one can identify the diamond associated to F F â„“ Fâ„“\mathscr{F} \ellFâ„“ with the minuscule Schubert cell defined by μ μ mu\muμ inside the B d R + B d R + B_(dR)^(+)B_{\mathrm{dR}}^{+}BdR+-Grassmannian for G G GGG. Once the map to Bun G Bun G Bun_(G)\operatorname{Bun}_{G}BunG is in the picture, one uses Fargues's result that the points of Bun G G _(G){ }_{G}G are in bijection with the Kottwitz set B ( G ) B ( G ) B(G)B(G)B(G), cf. [27] (see also [2] for an alternative proof that also works in equal characteristic). Moreover, the Newton decomposition is a stratification, in the sense that, for b B ( G , μ ) b ∈ B ( G , μ ) b in B(G,mu)b \in B(G, \mu)b∈B(G,μ), we have
F b ¯ = b b F b F â„“ b ¯ = ⨆ b ′ ≥ b   F â„“ b ′ bar(Fâ„“^(b))=⨆_(b^(') >= b)Fâ„“^(b^('))\overline{\mathscr{F} \ell^{b}}=\bigsqcup_{b^{\prime} \geq b} \mathscr{F} \ell^{b^{\prime}}Fâ„“b¯=⨆b′≥bFâ„“b′
where ≥ >=\geq≥ denotes the Bruhat order. The latter fact follows from a recent result of Viehmann, see [63, THEOREM 1.1].
On rank one points, π H T Ï€ H T pi_(HT)\pi_{\mathrm{HT}}Ï€HT is compatible with the two different ways of defining the Newton stratification: via pullback from S ¯ K p K p 0 S ¯ K p K p 0 bar(S)_(K^(p)K_(p)^(0))\bar{S}_{K^{p} K_{p}^{0}}S¯KpKp0 on § K p § K p §_(K^(p))\S_{K^{p}}§Kp and via pullback from Bun G G _(G){ }_{G}G on F F â„“ Fâ„“\mathscr{F} \ellFâ„“. The behavior is more subtle on higher rank points. This is related to the fact that the closure relations are reversed in the two settings: the basic locus inside S ¯ K p K p 0 S ¯ K p K p 0 bar(S)_(K^(p)K_(p)^(0))\bar{S}_{K^{p} K_{p}^{0}}S¯KpKp0 is the unique closed stratum, whereas each basic stratum inside B u n G B u n G Bun_(G)\mathrm{Bun}_{G}BunG is open. On the other hand, the ( μ ) ( μ ) (mu)(\mu)(μ)-ordinary locus is open and dense inside S ¯ K p K p 0 S ¯ K p K p 0 bar(S)_(K^(p)K_(p)^(0))\bar{S}_{K^{p} K_{p}^{0}}S¯KpKp0, whereas it is a zero-dimensional closed stratum inside
F F â„“ Fâ„“\mathscr{F} \ellFâ„“. The infinite-level product formula is established in [17, $4], although it is formulated in terms of functors on Perf E ˘ p 4 Perf E ˘ p â‹… 4 Perf_(E^(˘)_(p))*^(4)\operatorname{Perf}_{\breve{E}_{\mathfrak{p}}} \cdot{ }^{4}PerfE˘pâ‹…4 This was extended to Shimura varieties of Hodge type by Hamacher [31].
Assume that the Shimura varieties S K S K S_(K)S_{K}SK are compact. We have the following consequence for the fibers of π H T Ï€ H T pi_(HT)\pi_{\mathrm{HT}}Ï€HT : let x ¯ : S p a ( C , C + ) F b x ¯ : S p a C , C + → F â„“ b bar(x):Spa(C,C^(+))rarrFâ„“^(b)\bar{x}: \mathrm{Spa}\left(C, C^{+}\right) \rightarrow \mathscr{F} \ell^{b}x¯:Spa(C,C+)→Fâ„“b be a geometric point. Then there is an inclusion of g b â„‘ g b â„‘g^(b)\mathfrak{\Im g}^{b}â„‘gb into π H T 1 ( x ¯ ) Ï€ H T − 1 ( x ¯ ) pi_(HT)^(-1)( bar(x))\pi_{\mathrm{HT}}^{-1}(\bar{x})Ï€HT−1(x¯), which identifies the target with the canonical compactification of the source, in the sense of [54, PROPOSITION 18.6]. In [18, THEOREM 1.10], we extend the computation of the fibers to minimal and toroidal compactifications of (noncompact) Shimura varieties attached to quasisplit unitary groups. In this case, the fibers can be obtained from partial minimal and toroidal compactifications of Igusa varieties. It would be interesting to extend the whole infinite-level product formula to compactifications.
Example 3.4. We make the geometry of π H T Ï€ H T pi_(HT)\pi_{\mathrm{HT}}Ï€HT explicit in the case of the modular curve, i.e., for G = G L 2 / Q G = G L 2 / Q G=GL_(2)//QG=\mathrm{GL}_{2} / \mathbb{Q}G=GL2/Q. In this case, we identify F = P 1 , ad F â„“ = P 1 ,  ad  Fâ„“=P^(1," ad ")\mathscr{F} \ell=\mathbb{P}^{1, \text { ad }}Fâ„“=P1, ad  and we have the decomposition into Newton strata
The ordinary locus inside P 1 , ad P 1 ,  ad  P^(1," ad ")\mathbb{P}^{1, \text { ad }}P1, ad  consists of the set of points defined over Q p Q p Q_(p)\mathbb{Q}_{p}Qp and the basic / supersingular locus is its complement Ω Î© Omega\OmegaΩ, the Drinfeld upper half-plane.
The fibers of π H T Ï€ H T pi_(HT)\pi_{\mathrm{HT}}Ï€HT over the ordinary locus are "perfectoid versions" of Igusa curves. The infinite-level version of the product formula reduces, in this case, to the statement that the ordinary locus is parabolically induced from g ord â„‘ g ord  â„‘g^("ord ")\mathfrak{\Im} \mathrm{g}^{\text {ord }}â„‘gord , as in [ 19 , $ 6 ] [ 19 , $ 6 ] [19,$6][19, \$ 6][19,$6]. The fibers of π H T Ï€ H T pi_(HT)\pi_{\mathrm{HT}}Ï€HT over the supersingular locus are profinite sets: the corresponding Igusa varieties can be identified with double cosets D × D × ( A f p ) / K p D × ∖ D × A f p / K p D^(xx)\\D^(xx)(A_(f)^(p))//K^(p)D^{\times} \backslash D^{\times}\left(\mathbb{A}_{f}^{p}\right) / K^{p}D×∖D×(Afp)/Kp, where D / Q D / Q D//QD / \mathbb{Q}D/Q is the quaternion algebra ramified precisely at ∞ oo\infty∞ and p p ppp. This precise result is established in [35], although the idea goes back to DeuringSerre. One should be able to give an analogous description for basic Igusa varieties in much greater generality - this is closely related to Rapoport-Zink uniformization.

4. COHOMOLOGY WITH MOD â„“ â„“\ellâ„“ COEFFICIENTS

In this section, we outline some recent strategies for computing the cohomology of Shimura varieties with modulo â„“ â„“\ellâ„“ coefficients using the p p ppp-adic Hodge-Tate period morphism, where â„“ â„“\ellâ„“ and p p ppp are two distinct primes. We emphasize the strategies developed in [17-19,38].
We will assume throughout that ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is a Shimura datum of abelian type and, in practice, we will focus on two examples: the case of Shimura varieties associated with unitary similitude groups and the case of Hilbert modular varieties. Let m T m ⊂ T msubT\mathfrak{m} \subset \mathbb{T}m⊂T be a max- imal ideal in the support of H ( c ) ( S K ( C ) , F ) H ( c ) ∗ S K ( C ) , F â„“ H_((c))^(**)(S_(K)(C),F_(â„“))H_{(c)}^{*}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)H(c)∗(SK(C),Fâ„“). By work of Scholze (cf. [53, THEOREM 4.3.1]) and by the construction of Galois representations in the essentially self-dual case, we know in many cases how to associate a global modulo â„“ â„“\ellâ„“ Galois representation ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m to the maximal ideal m m m\mathfrak{m}m. Therefore, the non-Eisenstein condition makes sense, and one can at least formulate Conjecture 2.2. In order to make progress on this conjecture, we impose a local representation-theoretic condition at the prime p p ppp, which we treat as an auxiliary prime.
Definition 4.1. Let F F F\mathbb{F}F be a finite field of characteristic â„“ â„“\ellâ„“.
(1) Let p p ≠ â„“ p!=â„“p \neq \ellp≠ℓ be a prime, K / Q p K / Q p K//Q_(p)K / \mathbb{Q}_{p}K/Qp be a finite extension, and ρ ¯ : Gal ( K ¯ / K ) G L n ( F ) ρ ¯ : Gal ⁡ ( K ¯ / K ) → G L n ( F ) bar(rho):Gal( bar(K)//K)rarrGL_(n)(F)\bar{\rho}: \operatorname{Gal}(\bar{K} / K) \rightarrow \mathrm{GL}_{n}(\mathbb{F})ρ¯:Gal⁡(K¯/K)→GLn(F) be a continuous representation. We say that ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ is generic if it is unramified and the eigenvalues (with multiplicity) α 1 , , α n F ¯ α 1 , … , α n ∈ F ¯ â„“ alpha_(1),dots,alpha_(n)in bar(F)_(â„“)\alpha_{1}, \ldots, \alpha_{n} \in \overline{\mathbb{F}}_{\ell}α1,…,αn∈F¯ℓ of ρ ¯ ( Frob K ) ρ ¯ Frob K bar(rho)(Frob_(K))\bar{\rho}\left(\operatorname{Frob}_{K}\right)ρ¯(FrobK) satisfy α i / α j α i / α j ≠ alpha_(i)//alpha_(j)!=\alpha_{i} / \alpha_{j} \neqαi/αj≠ | O K / m K | O K / m K |O_(K)//m_(K)|\left|\mathcal{O}_{K} / \mathfrak{m}_{K}\right||OK/mK| for i j i ≠ j i!=ji \neq ji≠j
(2) Let F F FFF be a number field and ρ ¯ : Gal ( F ¯ / F ) G L n ( F ) ρ ¯ : Gal ⁡ ( F ¯ / F ) → G L n ( F ) bar(rho):Gal( bar(F)//F)rarrGL_(n)(F)\bar{\rho}: \operatorname{Gal}(\bar{F} / F) \rightarrow \mathrm{GL}_{n}(\mathbb{F})ρ¯:Gal⁡(F¯/F)→GLn(F) be a continuous representation. We say that a prime p p ≠ â„“ p!=â„“p \neq \ellp≠ℓ is decomposed generic for ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ if p p ppp splits completely in F F FFF and, for every prime p p p ∣ p p∣p\mathfrak{p} \mid pp∣p of F , ρ ¯ | Gal ( F ¯ p / F p ) F , ρ ¯ Gal ⁡ F ¯ p / F p F,( bar(rho))|_(Gal( bar(F)_(p)//F_(p)))F,\left.\bar{\rho}\right|_{\operatorname{Gal}\left(\bar{F}_{\mathfrak{p}} / F_{\mathfrak{p}}\right)}F,ρ¯|Gal⁡(F¯p/Fp) is generic. We say that ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ is decomposed generic if there exists a prime p p ≠ â„“ p!=â„“p \neq \ellp≠ℓ which is decomposed generic for ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯. (If one such prime exists, then infinitely many do.)
Remark 4.2. The condition for the local representation ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ of G a l ( K ¯ / K ) G a l ( K ¯ / K ) Gal( bar(K)//K)\mathrm{Gal}(\bar{K} / K)Gal(K¯/K) to be generic implies that any lift to characteristic 0 of ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ corresponds under the local Langlands correspondence to a generic principal series representation of G L n ( K ) G L n ( K ) GL_(n)(K)\mathrm{GL}_{n}(K)GLn(K). Such a representation can never arise from a nonsplit inner form of G L n / K G L n / K GL_(n)//K\mathrm{GL}_{n} / KGLn/K via the Jacquet-Langlands correspondence. For this reason, a generic ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ cannot be the modulo â„“ â„“\ellâ„“ reduction of the L L LLL-parameter of a smooth representation of a nonsplit inner form of G L n / K G L n / K GL_(n)//K\mathrm{GL}_{n} / KGLn/K.
A semisimple 2-dimensional representation ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ of Gal ( Q ¯ / Q ) Gal ⁡ ( Q ¯ / Q ) Gal( bar(Q)//Q)\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})Gal⁡(Q¯/Q) is either decomposed generic or it satisfies (2.1): the case where ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ is a direct sum of two characters can be analyzed by hand, and the case where ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ is absolutely irreducible follows from the paragraph after Theorem 3.1 in [37]. More generally, the condition for a global representation ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ of Gal ( F ¯ / F ) Gal ⁡ ( F ¯ / F ) Gal( bar(F)//F)\operatorname{Gal}(\bar{F} / F)Gal⁡(F¯/F) to be decomposed generic can be ensured when ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ has large image. For example, if > 2 , F â„“ > 2 , F â„“ > 2,F\ell>2, Fâ„“>2,F is a totally real field, and ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ is a totally odd 2-dimensional representation with nonsolvable image, then ρ ¯ ρ ¯ bar(rho)\bar{\rho}ρ¯ is decomposed generic (cf. [19, LEMMA 7.1.8]).
Let F F FFF be an imaginary CM field. Let ( B , , V , , ) ( B , ∗ , V , ⟨ ⋅ , ⋅ ⟩ ) (B,**,V,(:*,*:))(B, *, V,\langle\cdot, \cdot\rangle)(B,∗,V,⟨⋅,⋅⟩) be a PEL datum of type A A AAA, where B B BBB is a central simple algebra with center F F FFF. We let ( G , X ) ( G , X ) (G,X)(G, X)(G,X) be the associated Shimura datum. For a neat compact open subgroup K G ( A f ) K ⊂ G A f K sub G(A_(f))K \subset G\left(\mathbb{A}_{f}\right)K⊂G(Af), we let S K / E S K / E S_(K)//ES_{K} / ESK/E be the associated Shimura variety, of dimension d d ddd. The following conjecture is a slightly different formulation of [38, CONJECTURE 1.2], with essentially the same content.
Conjecture 4.3. Let n T n ⊂ T nsubT\mathfrak{n} \subset \mathbb{T}n⊂T be a maximal ideal in the support of H ( c ) i ( S K ( C ) , F ) H ( c ) i S K ( C ) , F â„“ H_((c))^(i)(S_(K)(C),F_(â„“))H_{(c)}^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)H(c)i(SK(C),Fâ„“). Assume that ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m is decomposed generic. Then the following statements hold true:
(1) if H c i ( S K ( C ) , F ) m 0 H c i S K ( C ) , F ℓ m ≠ 0 H_(c)^(i)(S_(K)(C),F_(ℓ))_(m)!=0H_{c}^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}} \neq 0Hci(SK(C),Fℓ)m≠0, then i d i ≤ d i <= di \leq di≤d;
(2) if H i ( S K ( C ) , F ) m 0 H i S K ( C ) , F ℓ m ≠ 0 H^(i)(S_(K)(C),F_(ℓ))_(m)!=0H^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}} \neq 0Hi(SK(C),Fℓ)m≠0, then i d i ≥ d i >= di \geq di≥d.
If the Shimura varieties S K S K S_(K)S_{K}SK are compact, or if we additionally assume m m m\mathfrak{m}m to be nonEisenstein, Conjecture 4.3 implies a significant part of Conjecture 2.2 for Shimura varieties of PEL type A. Analogues of Conjecture 4.3 can be formulated (and are perhaps within reach) for other Shimura varieties, such as Siegel modular varieties.
Theorem 4.4 ([17] strengthened in [38]). Assume that G G GGG is anisotropic modulo center, so that the Shimura varieties S K S K S_(K)S_{K}SK are compact. Then Conjecture 4.3 holds true.
Theorem 4.5 ([18] strengthened in [38]). Assume that B = F , V = F 2 n B = F , V = F 2 n B=F,V=F^(2n)B=F, V=F^{2 n}B=F,V=F2n and G G GGG is a quasisplit group of unitary similitudes. Then Conjecture 4.3 holds true.
Remark 4.6. The more recent results of [38] have significantly fewer technical assumptions than the earlier ones of [17] and [18]. For example, Koshikawa's version of Theorem 4.5 allows F F FFF to be an imaginary quadratic field. It seems nontrivial to obtain this case with the methods of [18]. In the noncompact case, his results rely on the geometric constructions in [18], in particular on the semiperversity result for Shimura varieties attached to quasisplit unitary groups that is established there. As he notes, a generalization of this semiperversity result should lead to a full proof of Conjecture 4.3 for Shimura varieties of PEL type A. The more general semiperversity result will be obtained in the upcoming PhD thesis of Mafalda Santos.
In the case of Harris-Taylor Shimura varieties, Theorem 4.4 was first proved by Boyer [9]. Boyer's argument uses the integral models of Shimura varieties of Harris-Taylor type, but it is close in spirit to the argument carried out in [17] on the generic fiber. What is really interesting about Boyer's results is that he goes beyond genericity, in the following sense. Given the eigenvalues (with multiplicity) α 1 , , α n α 1 , … , α n alpha_(1),dots,alpha_(n)\alpha_{1}, \ldots, \alpha_{n}α1,…,αn of ρ ¯ m ( ρ ¯ m bar(rho)_(m)(:}\bar{\rho}_{\mathfrak{m}}\left(\right.ρ¯m( Frob p ) p {:_(p))\left._{\mathfrak{p}}\right)p), with p p p ∣ p p∣p\mathfrak{p} \mid pp∣p the relevant prime of F , 5 F , 5 F,^(5)F,{ }^{5}F,5 one can define a "defect" that measures how far ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m is from being generic at p p p\mathfrak{p}p. Concretely, set δ p ( m ) δ p ( m ) delta_(p)(m)\delta_{\mathfrak{p}}(\mathfrak{m})δp(m) to be equal to the length of the maximal chain of eigenvalues where the successive terms have ratio equal to | O F p / m F p | O F p / m F p |O_(F_(p))//m_(F_(p))|\left|\mathcal{O}_{F_{\mathfrak{p}}} / \mathfrak{m}_{F_{\mathfrak{p}}}\right||OFp/mFp|. Boyer shows that the cohomology groups H ( c ) i ( S K ( C ) , F ) m H ( c ) i S K ( C ) , F â„“ m H_((c))^(i)(S_(K)(C),F_(â„“))_(m)H_{(c)}^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)i(SK(C),Fâ„“)m are nonzero at most in the range [ d δ p ( m ) , d + δ p ( m ) ] d − δ p ( m ) , d + δ p ( m ) [d-delta_(p)(m),d+delta_(p)(m)]\left[d-\delta_{\mathfrak{p}}(\mathfrak{m}), d+\delta_{\mathfrak{p}}(\mathfrak{m})\right][d−δp(m),d+δp(m)]. As noted by both Emerton and Koshikawa, such a result is consistent with Arthur's conjectures on the cohomology of Shimura varieties with C C C\mathbb{C}C-coefficients and points towards a modulo â„“ â„“\ellâ„“ analogue of these conjectures.
Let us also discuss the analogous vanishing result in the Hilbert case. Let F F FFF be a totally real field of degree g g ggg and let G = Res F / Q G L 2 G = Res F / Q ⁡ G L 2 G=Res_(F//Q)GL_(2)G=\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{2}G=ResF/Q⁡GL2. For a neat compact open subgroup K G ( A f ) K ⊂ G A f K sub G(A_(f))K \subset G\left(\mathbb{A}_{f}\right)K⊂G(Af), we let S K / Q S K / Q S_(K)//QS_{K} / \mathbb{Q}SK/Q be the corresponding Hilbert modular variety, of dimension g g ggg.
Theorem 4.7 ([19, THEOREM A]). Let > 2 â„“ > 2 â„“ > 2\ell>2â„“>2 and m T m ⊂ T msubT\mathfrak{m} \subset \mathbb{T}m⊂T be a maximal ideal in the support of H ( c ) i ( S K ( C ) , F ) H ( c ) i S K ( C ) , F â„“ H_((c))^(i)(S_(K)(C),F_(â„“))H_{(c)}^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)H(c)i(SK(C),Fâ„“). Assume that the image of ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m is not solvable, which implies that ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m is absolutely irreducible and decomposed generic. Then H c i ( S K ( C ) , F ) m = H i ( S K ( C ) , F ) m H c i S K ( C ) , F â„“ m = H i S K ( C ) , F â„“ m H_(c)^(i)(S_(K)(C),F_(â„“))_(m)=H^(i)(S_(K)(C),F_(â„“))_(m)H_{c}^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}}=H^{i}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}}Hci(SK(C),Fâ„“)m=Hi(SK(C),Fâ„“)m is nonzero only for i = g i = g i=gi=gi=g.
5 In this special case, one does not have to impose the condition that p p ppp splits completely in F F FFF, and it suffices to have genericity at one prime p p p ∣ p p∣p\mathfrak{p} \mid pp∣p.
The same result holds for all quaternionic Shimura varieties, and we can even prove the analogue of Boyer's result that goes beyond genericity in all these settings. As an application, we deduce (under some technical assumptions) that the p p ppp-adic local Langlands correspondence for G L 2 ( Q p ) G L 2 Q p GL_(2)(Q_(p))\mathrm{GL}_{2}\left(\mathbb{Q}_{p}\right)GL2(Qp) occurs in the completed cohomology of Hilbert modular varieties, when p p ppp is a prime that splits completely in F F FFF. This uses the axiomatic approach via patching introduced in [14] and further developed in [15,30].
We now outline the original strategy for proving Theorem 4.4, which was introduced in [17]. Let p p ppp be a prime and K = K p K p G ( A f ) K = K p K p ⊂ G A f K=K^(p)K_(p)sub G(A_(f))K=K^{p} K_{p} \subset G\left(\mathbb{A}_{f}\right)K=KpKp⊂G(Af) be a neat compact open subgroup. The Hodge-Tate period morphism gives rise to a T T T\mathbb{T}T-equivariant diagram
The standard comparison theorems between various cohomology theories allow us to identify H ( c ) ( S K ( C ) , F ) m H ( c ) ∗ S K ( C ) , F â„“ m H_((c))^(**)(S_(K)(C),F_(â„“))_(m)H_{(c)}^{*}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)∗(SK(C),Fâ„“)m with H ( c ) ( S K , F ) m H ( c ) ∗ S K , F â„“ m H_((c))^(**)(S_(K),F_(â„“))_(m)H_{(c)}^{*}\left(S_{K}, \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)∗(SK,Fâ„“)m. The arrow on the left-hand side of (4.1) is a K p K p K_(p)K_{p}Kp-torsor, so the Hochschild-Serre spectral sequence allows us to recover H ( c ) ( S K , F ) m H ( c ) ∗ S K , F â„“ m H_((c))^(**)(S_(K),F_(â„“))_(m)H_{(c)}^{*}\left(S_{K}, \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)∗(SK,Fâ„“)m from H ( c ) ( ς K p , F ) m H ( c ) ∗ Ï‚ K p , F â„“ m H_((c))^(**)(Ï‚_(K^(p)),F_(â„“))_(m)H_{(c)}^{*}\left(\varsigma_{K^{p}}, \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)∗(Ï‚Kp,Fâ„“)m. The idea is now to compute H ( c ) ( ς K p , F ) m H ( c ) ∗ Ï‚ K p , F â„“ m H_((c))^(**)(Ï‚_(K^(p)),F_(â„“))_(m)H_{(c)}^{*}\left(\varsigma_{K^{p}}, \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)∗(Ï‚Kp,Fâ„“)m in two stages: first understand the complex of sheaves ( R π H T F ) m R Ï€ H T ∗ F â„“ m (Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m on F F â„“ Fâ„“\mathscr{F} \ellFâ„“, then compute the total cohomology using the Leray-Serre spectral sequence.
Two miraculous things happen that greatly simplify the structure of ( R π H T F ) m R Ï€ H T ∗ F â„“ m (Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m. The first is that ( R π H T F ) m R Ï€ H T ∗ F â„“ m (Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m behaves like a perverse sheaf on F F â„“ Fâ„“\mathscr{F} \ellFâ„“. This is because π H T Ï€ H T pi_(HT)\pi_{\mathrm{HT}}Ï€HT is simultaneously affinoid, as discussed after Theorem 3.1, and partially proper, because the Shimura varieties were assumed to be compact. In particular, the restriction of ( R π H T F ) m R Ï€ H T ∗ F â„“ m (Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m to a highest-dimensional stratum in its support is concentrated in one degree. By the computation of the fibers of π H T Ï€ H T pi_(HT)\pi_{\mathrm{HT}}Ï€HT, this implies that the cohomology groups R Γ ( g b , Z ) m R Γ â„‘ g b , Z â„“ m R Gamma((â„‘g)^(b),Z_(â„“))_(m)R \Gamma\left(\mathfrak{\Im g}{ }^{b}, \mathbb{Z}_{\ell}\right)_{\mathfrak{m}}RΓ(â„‘gb,Zâ„“)m are concentrated in one degree and torsion-free. The second miracle is that, whenever the group G b ( Q p ) G b Q p G_(b)(Q_(p))G_{b}\left(\mathbb{Q}_{p}\right)Gb(Qp) acting on s g b s g b sg^(b)\mathfrak{s g}^{b}sgb comes from a nonquasisplit inner form, the localization R Γ ( s g b , Q ) m R Γ s g b , Q â„“ m R Gamma(sg^(b),Q_(â„“))_(m)R \Gamma\left(\mathfrak{s g}^{b}, \mathbb{Q}_{\ell}\right)_{\mathfrak{m}}RΓ(sgb,Qâ„“)m vanishes. This uses the genericity of ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m at each p p p ∣ p p∣p\mathfrak{p} \mid pp∣p and suggests that the cohomology of Igusa varieties satisfies some form of local-global compatibility. Finally, the condition that p p ppp splits completely in F F FFF guarantees that the only Newton stratum for which G b G b G_(b)G_{b}Gb is quasisplit is the ordinary one. Therefore, the hypotheses of Theorem 4.4 guarantee that ( R π H T F ) m R Ï€ H T ∗ F â„“ m (Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m is as simple as possible - it is supported in one degree on a zero-dimensional stratum!
The computation of R Γ ( I g b , Q ) m R Γ I g b , Q â„“ m R Gamma(Ig^(b),Q_(â„“))_(m)R \Gamma\left(\mathrm{Ig}^{b}, \mathbb{Q}_{\ell}\right)_{\mathfrak{m}}RΓ(Igb,Qâ„“)m, at least at the level of the Grothendieck group, can be done using the trace formula method pioneered by Shin [60]. This is the method used for Shimura varieties of PEL type A in [17] and [18]. For inner forms of Res F / Q G L 2 Res F / Q ⁡ G L 2 Res_(F//Q)GL_(2)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{2}ResF/Q⁡GL2, with F F FFF a totally real field, one can avoid these difficult computations, cf. [19]. In this setting, one can reinterpret results of Tian-Xiao [62] on geometric instances of the Jacquet-Langlands correspondence as giving rise to exotic isomorphisms between Igusa varieties arising from different Shimura varieties. This is what happens for the basic stratum in Example 3.4. Then one can conclude by applying the classical Jacquet-Langlands correspondence.
In [38], Koshikawa introduces a novel and complementary strategy for proving these kinds of vanishing theorems. He shows that, under the same genericity assumption in Definition 4.1, only the restriction of ( R π H T F ) m R Ï€ H T ∗ F â„“ m (Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m to the ordinary locus contributes to the total cohomology of the Shimura variety. To achieve this, he proves the analogous generic vanishing theorem for the cohomology R Γ c ( M b , Z ) m p R Γ c M b , Z â„“ m p RGamma_(c)(M^(b),Z_(â„“))_(m_(p))R \Gamma_{c}\left(\mathcal{M}^{b}, \mathbb{Z}_{\ell}\right)_{\mathfrak{m}_{p}}RΓc(Mb,Zâ„“)mp of the Rapoport-Zink space, where m p m p m_(p)\mathfrak{m}_{p}mp is a maximal ideal of the local spherical Hecke algebra at p p ppp. This relies on the recent work of Fargues-Scholze on the geometrization of the local Langlands conjecture [28].
Koshikawa's strategy is more flexible, allowing him to handle with ease the case where F F FFF is an imaginary quadratic field. On the other hand, the original approach also gives information about the complexes of sheaves ( R π H T F ) m R Ï€ H T ∗ F â„“ m (Rpi_(HT**)F_(â„“))_(m)\left(R \pi_{\mathrm{HT} *} \mathbb{F}_{\ell}\right)_{\mathfrak{m}}(RÏ€HT∗Fâ„“)m, rather than just about the cohomology groups H ( c ) ( S K ( C ) , F ) m H ( c ) ∗ S K ( C ) , F â„“ m H_((c))^(**)(S_(K)(C),F_(â„“))_(m)H_{(c)}^{*}\left(S_{K}(\mathbb{C}), \mathbb{F}_{\ell}\right)_{\mathfrak{m}}H(c)∗(SK(C),Fâ„“)m. These complexes should play an important role for questions of local-global compatibility in Fargues's geometrization conjecture, cf. [26, 87].

5. COHOMOLOGY WITH MOD p p ppp AND p p ppp-ADIC COEFFICIENTS

The most general method for constructing p p ppp-adic families of automorphic forms from the cohomology of locally symmetric spaces is via completed cohomology. First introduced by Emerton in [24], this has the following definition:
H ~ ( K p , Z p ) = lim n ( lim ( l i m K p ( X K p K p , Z / p n ) ) H ~ ∗ K p , Z p = lim n   lim ∗   ( l i m K p X K p K p , Z / p n {: tilde(H)^(**)(K^(p),Z_(p))=lim _(n)(lim**_((lim_(K_(p)))(X_(K^(p)K_(p)),Z//p^(n)))\left.\tilde{H}^{*}\left(K^{p}, \mathbb{Z}_{p}\right)=\underset{n}{\lim } \underset{\underset{K_{p}}{(l i m}}{\left(\lim ^{*}\right.}\left(X_{K^{p} K_{p}}, \mathbb{Z} / p^{n}\right)\right)H~∗(Kp,Zp)=limn(lim∗(limKp(XKpKp,Z/pn))
where K p G ( A f ) K p ⊂ G A f K^(p)sub G(A_(f))K^{p} \subset G\left(\mathbb{A}_{f}\right)Kp⊂G(Af) is a sufficiently small, fixed tame level, and K p G ( Q p ) K p ⊂ G Q p K_(p)sub G(Q_(p))K_{p} \subset G\left(\mathbb{Q}_{p}\right)Kp⊂G(Qp) runs over compact open subgroups. This space has an action of the spherical Hecke algebra T T T\mathbb{T}T, built from Hecke operators away from p p ppp, as well as an action of the group G ( Q p ) G Q p G(Q_(p))G\left(\mathbb{Q}_{p}\right)G(Qp). One can make the analogous definition for completed cohomology with compact support, and a variant gives completed homology and completed Borel-Moore homology. See [25] for an excellent survey that gives motivation, examples, and sketches the basic properties of these spaces.
Motivated by heuristics from the p p ppp-adic Langlands programme, Calegari and Emerton made several conjectures about the range of degrees in which one can have nonzero completed (co)homology and about the codimension of completed homology over the completed group rings Z p [ [ K p ] ] Z p [ [ K p ] ] Z_(p)[[K_(p)]]\mathbb{Z}_{p} \llbracket K_{p} \rrbracketZp[[Kp]]. See [11, coNJECTURE 1.5] for the original formulation and [32, CONJECTURE 1.3] for a slightly different formulation, which emphasizes the natural implications between the various conjectures. In particular, Calegari-Emerton conjectured that
H ~ c i ( K p , Z p ) = H ~ i ( K p , Z p ) = 0 for i > q 0 H ~ c i K p , Z p = H ~ i K p , Z p = 0  for  i > q 0 tilde(H)_(c)^(i)(K^(p),Z_(p))= tilde(H)^(i)(K^(p),Z_(p))=0quad" for "i > q_(0)\tilde{H}_{c}^{i}\left(K^{p}, \mathbb{Z}_{p}\right)=\tilde{H}^{i}\left(K^{p}, \mathbb{Z}_{p}\right)=0 \quad \text { for } i>q_{0}H~ci(Kp,Zp)=H~i(Kp,Zp)=0 for i>q0
For Shimura varieties of preabelian type, the Calegari-Emerton conjectures were proved by Hansen-Johansson in [32], building on work of Scholze who proved the vanishing of completed cohomology with compact support for Shimura varieties of Hodge type [53].
We sketch Scholze's argument, which illustrates the role of p p ppp-adic geometry in this result. It is enough to show that
H ~ c i ( K p , F p ) = lim K p H c i ( S K p K p ( C ) , F p ) H ~ c i K p , F p = lim K p   H c i S K p K p ( C ) , F p tilde(H)_(c)^(i)(K^(p),F_(p))=lim_(K_(p))H_(c)^(i)(S_(K^(p)K_(p))(C),F_(p))\tilde{H}_{c}^{i}\left(K^{p}, \mathbb{F}_{p}\right)=\underset{K_{p}}{\lim } H_{c}^{i}\left(S_{K^{p} K_{p}}(\mathbb{C}), \mathbb{F}_{p}\right)H~ci(Kp,Fp)=limKpHci(SKpKp(C),Fp)
vanishes for i > d = dim E S K i > d = dim E ⁡ S K i > d=dim_(E)S_(K)i>d=\operatorname{dim}_{E} S_{K}i>d=dimE⁡SK. Since ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is a Shimura datum of Hodge type, we are in the setting of Theorem 3.1 - in fact, we know that the minimal compactification S K p S K p ∗ S_(K^(p))^(**)S_{K^{p}}^{*}SKp∗ is perfectoid. The primitive comparison theorem of [52] gives an almost isomorphism between H ~ c i ( K p , F p ) O C / p H ~ c i K p , F p ⊗ O C / p tilde(H)_(c)^(i)(K^(p),F_(p))oxO_(C)//p\tilde{H}_{c}^{i}\left(K^{p}, \mathbb{F}_{p}\right) \otimes \mathcal{O}_{C} / pH~ci(Kp,Fp)⊗OC/p and H e t i ( S K p , d + / p ) H e t i S K p ∗ , d + / p H_(et)^(i)(S_(K^(p))^(**),d^(+)//p)H_{\mathrm{et}}^{i}\left(\mathcal{S}_{K^{p}}^{*}, d^{+} / p\right)Heti(SKp∗,d+/p), where + O + + ⊆ O + ^(+)subeO^(+)\mathscr{}^{+} \subseteq \mathcal{O}^{+}+⊆O+is the subsheaf of sections that vanish along the boundary. On an affinoid perfectoid space, Scholze proved the almost vanishing of the étale cohomology of O + / p O + / p O^(+)//p\mathcal{O}^{+} / pO+/p in degree i > 0 i > 0 i > 0i>0i>0. With some care at the boundary, one deduces that it is enough to prove that the analytic cohomology groups H a n i ( S K p , L + / p ) H a n i S K p ∗ , L + / p H_(an)^(i)(S_(K^(p))^(**),L^(+)//p)H_{\mathrm{an}}^{i}\left(\mathcal{S}_{K^{p}}^{*}, \mathcal{L}^{+} / p\right)Hani(SKp∗,L+/p) are almost 0 in degree i > d i > d i > di>di>d. This final step follows from a theorem of Scheiderer on the cohomological dimension of spectral spaces.
In [20] and [16], we study Shimura varieties with unipotent level at p p ppp. More precisely, assume that ( G , X ) ( G , X ) (G,X)(G, X)(G,X) is a Shimura datum of Hodge type and that G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp is split. Choose a split model of G G GGG and a Borel subgroup B B BBB over Z p Z p Z_(p)\mathbb{Z}_{p}Zp, and let U B U ⊂ B U sub BU \subset BU⊂B be the unipotent radical.
Theorem 5.1 ([16, THEOREM 1.1]). Let H U ( Z p ) H ⊆ U Z p H sube U(Z_(p))H \subseteq U\left(\mathbb{Z}_{p}\right)H⊆U(Zp) be a closed subgroup. We have
lim K p H H c i ( S K p K p ( C ) , F p ) = 0 for i > d lim K p ⊇ H   H c i S K p K p ( C ) , F p = 0  for  i > d lim_(K_(p)supe H)H_(c)^(i)(S_(K^(p)K_(p))(C),F_(p))=0quad" for "i > d\underset{K_{p} \supseteq H}{\lim } H_{c}^{i}\left(S_{K^{p} K_{p}}(\mathbb{C}), \mathbb{F}_{p}\right)=0 \quad \text { for } i>dlimKp⊇HHci(SKpKp(C),Fp)=0 for i>d
This result is stronger than the Calegari-Emerton conjecture for completed cohomology with compact support, since we can take H = { 1 } H = { 1 } H={1}H=\{1\}H={1} and recover Scholze's result discussed above. In addition to the argument sketched above, the key new idea needed for Theorem 5.1 is that the Bruhat decomposition on the Hodge-Tate period domain F F ℓ Fℓ\mathscr{F} \ellFℓ remembers how far different subspaces of S K p U ( Z p ) S K p U Z p ∗ S_(K^(p)U(Z_(p)))^(**)\mathcal{S}_{K^{p} U\left(\mathbb{Z}_{p}\right)}^{*}SKpU(Zp)∗ are from being perfectoid.
Example 5.2. Assume that G = G L 2 / Q G = G L 2 / Q G=GL_(2)//QG=\mathrm{GL}_{2} / \mathbb{Q}G=GL2/Q, so that we are working in the modular curve case. The Bruhat decomposition is given by P 1 , ad = A 1 , ad { } P 1 ,  ad  = A 1 ,  ad  ⊔ { ∞ } P^(1," ad ")=A^(1," ad ")⊔{oo}\mathbb{P}^{1, \text { ad }}=\mathbb{A}^{1, \text { ad }} \sqcup\{\infty\}P1, ad =A1, ad ⊔{∞}, with the two Bruhat cells in natural bijection with the two components of the ordinary locus in (3.5). We have a morphism of sites
π H T / U ( Z p ) : ( S K U ( Z p ) ) ét | P 1 , a d | / U ( Z p ) Ï€ H T / U Z p : S K ∗ U Z p ét  → P 1 , a d / U Z p pi_(HT//U(Z_(p))):(S_(K)^(**)U(Z_(p)))_("ét ")rarr|P^(1,ad)|//U(Z_(p))\pi_{\mathrm{HT} / U\left(\mathbb{Z}_{p}\right)}:\left(\mathcal{S}_{K}^{*} U\left(\mathbb{Z}_{p}\right)\right)_{\text {ét }} \rightarrow\left|\mathbb{P}^{1, \mathrm{ad}}\right| / U\left(\mathbb{Z}_{p}\right)Ï€HT/U(Zp):(SK∗U(Zp))ét →|P1,ad|/U(Zp)
where we take the quotient | P 1 , a d | / U ( Z p ) P 1 , a d / U Z p |P^(1,ad)|//U(Z_(p))\left|\mathbb{P}^{1, \mathrm{ad}}\right| / U\left(\mathbb{Z}_{p}\right)|P1,ad|/U(Zp) only as a topological space. The preimage of | A 1 , ad | / U ( Z p ) A 1 ,  ad  / U Z p |A^(1," ad ")|//U(Z_(p))\left|\mathbb{A}^{1, \text { ad }}\right| / U\left(\mathbb{Z}_{p}\right)|A1, ad |/U(Zp) in S K p U ( Z p ) S K p U Z p S_(K^(p)U(Z_(p)))S_{K^{p} U\left(\mathbb{Z}_{p}\right)}SKpU(Zp) is a perfectoid space, as proved by Ludwig in [42]. The preimage of | | / U ( Z p ) | ∞ | / U Z p |oo|//U(Z_(p))|\infty| / U\left(\mathbb{Z}_{p}\right)|∞|/U(Zp) has a Z p Z p Z_(p)\mathbb{Z}_{p}Zp-cover that is an affinoid perfectoid space. This allows us to bound the support of each R i π H T / U ( Z p ) ( L + / p ) R i Ï€ H T ∗ / U Z p L + / p R^(i)pi_(HT**//U(Z_(p)))(L^(+)//p)R^{i} \pi_{\mathrm{HT} * / U\left(\mathbb{Z}_{p}\right)}\left(\mathscr{L}^{+} / p\right)RiÏ€HT∗/U(Zp)(L+/p), and we conclude by the Leray spectral sequence.
More generally, the Bruhat decomposition G = w W P μ B w P μ G = ⨆ w ∈ W P μ   B w P μ G=⨆_(w inW^(P mu))BwP_(mu)G=\bigsqcup_{w \in W^{P \mu}} B w P_{\mu}G=⨆w∈WPμBwPμ gives a decomposition F = w W P μ F w F â„“ = ⨆ w ∈ W P μ   F â„“ w Fâ„“=⨆_(w inW^(P_(mu)))Fâ„“^(w)\mathscr{F} \ell=\bigsqcup_{w \in W^{P_{\mu}}} \mathscr{F} \ell^{w}Fâ„“=⨆w∈WPμFâ„“w into locally closed Schubert cells indexed by certain Weyl group elements. For each F w / U ( Z p ) F â„“ w / U Z p Fâ„“^(w)//U(Z_(p))\mathscr{F} \ell^{w} / U\left(\mathbb{Z}_{p}\right)Fâ„“w/U(Zp), we can quantify how far its preimage in S K p U ( Z p ) S K p U Z p ∗ S_(K^(p)U(Z_(p)))^(**)S_{K^{p} U\left(\mathbb{Z}_{p}\right)}^{*}SKpU(Zp)∗ is from being a perfectoid space, which depends on the length of the Weyl group element w w www. The assumption that G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp is split guarantees that all the Weyl group elements lie in the ordinary locus inside F F â„“ Fâ„“\mathscr{F} \ellFâ„“, which greatly simplifies the analysis. However, the analogue of Theorem 5.1 may hold even without the assumption that G Q p G Q p G_(Q_(p))G_{\mathbb{Q}_{p}}GQp is split, and even when the ordinary locus is empty. There is some evidence in this direction, e.g., by using embeddings into higher-dimensional Shimura varieties attached to split groups, or by using the results of [36] to handle the Harris-Taylor case, as in the upcoming PhD thesis of Louis Jaburi.
The Bruhat decomposition on F F â„“ Fâ„“\mathscr{F} \ellFâ„“ has more recently been used by Boxer and Pilloni to define a version of higher Coleman theory indexed by each w W P μ w ∈ W P μ w inW^(P_(mu))w \in W^{P_{\mu}}w∈WPμ in [8]. The development of higher Coleman and higher Hida theories shows that the geometric theory of p p ppp-adic automorphic forms on Shimura varieties is much richer than previously expected. Furthermore, the Bruhat decomposition indicates the form a p p ppp-adic Eichler-Shimura isomorphism should take, relating completed cohomology to these more geometric theories. In joint work in progress with Mantovan and Newton, we use the geometry described in Example 5.2 to give a new proof of the ordinary Eichler-Shimura isomorphism due to Ohta [46, 47]. Our result decomposes the ordinary completed cohomology of the modular curve in terms of Hida theory and higher Hida theory, the latter recently developed by Boxer and Pilloni in [7].
Theorem 5.1 seems far away from Conjecture 2.2, because it concerns Shimura varieties with "infinite level" at p p ppp. However, one could ask whether a version of Theorem 5.1 holds already at level B ( Z p ) B Z p B(Z_(p))B\left(\mathbb{Z}_{p}\right)B(Zp), at least after applying an ordinary idempotent, as in Hida theory. If that were the case, the control theorems in Hida theory (specifically the result known as independence of level) and a careful application of Poincaré duality would imply that an = p ℓ = p ℓ=p\ell=pℓ=p analogue of Conjecture 4.3 holds, with generic replaced by ordinary. More precisely, in this case, the "auxiliary prime" p p ppp where we impose a representation-theoretic condition is no longer auxiliary but rather equal to ℓ ℓ\ellℓ.

6. APPLICATIONS BEYOND SHIMURA VARIETIES

While the focus of this article has been the cohomology of Shimura varieties, Theorems 4.5 and 5.1 have surprising applications to understanding the cohomology of more general locally symmetric spaces. For example, let F F FFF be an imaginary C M C M CM\mathrm{CM}CM field and G = G = G=G=G= Res F / Q G L n Res F / Q ⁡ G L n Res_(F//Q)GL_(n)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GL}_{n}ResF/Q⁡GLn. Then G G GGG can be realized as the Levi quotient of the Siegel maximal parabolic of a quasisplit unitary group G ~ G ~ tilde(G)\tilde{G}G~. The Borel-Serre compactification X ~ K ~ B S X ~ K ~ B S tilde(X)_( tilde(K))^(BS)\tilde{X}_{\tilde{K}}^{\mathrm{BS}}X~K~BS for the locally symmetric spaces associated with the unitary group G ~ G ~ tilde(G)\tilde{G}G~ gives rise to a Hecke-equivariant long exact sequence of the form
H c i ( X ~ K ~ , Z / n Z ) H i ( X ~ K ~ , Z / n Z ) H i ( X ~ K ~ , Z / n Z ) (6.1) H c i + 1 ( X ~ K ~ , Z / n Z ) ⋯ → H c i X ~ K ~ , Z / ℓ n Z → H i X ~ K ~ , Z / ℓ n Z → H i ∂ X ~ K ~ , Z / ℓ n Z (6.1) → H c i + 1 X ~ K ~ , Z / ℓ n Z → ⋯ {:[cdots rarrH_(c)^(i)( tilde(X)_( tilde(K)),Z//ℓ^(n)Z)rarrH^(i)( tilde(X)_( tilde(K)),Z//ℓ^(n)Z)rarrH^(i)(del tilde(X)_( tilde(K)),Z//ℓ^(n)Z)],[(6.1) rarrH_(c)^(i+1)( tilde(X)_( tilde(K)),Z//ℓ^(n)Z)rarr cdots]:}\begin{align*} \cdots & \rightarrow H_{c}^{i}\left(\tilde{X}_{\tilde{K}}, \mathbb{Z} / \ell^{n} \mathbb{Z}\right) \rightarrow H^{i}\left(\tilde{X}_{\tilde{K}}, \mathbb{Z} / \ell^{n} \mathbb{Z}\right) \rightarrow H^{i}\left(\partial \tilde{X}_{\tilde{K}}, \mathbb{Z} / \ell^{n} \mathbb{Z}\right) \\ & \rightarrow H_{c}^{i+1}\left(\tilde{X}_{\tilde{K}}, \mathbb{Z} / \ell^{n} \mathbb{Z}\right) \rightarrow \cdots \tag{6.1} \end{align*}⋯→Hci(X~K~,Z/ℓnZ)→Hi(X~K~,Z/ℓnZ)→Hi(∂X~K~,Z/ℓnZ)(6.1)→Hci+1(X~K~,Z/ℓnZ)→⋯
where X ~ K ~ = X ~ K ~ B S X ~ K ~ ∂ X ~ K ~ = X ~ K ~ B S ∖ X ~ K ~ del tilde(X)_( tilde(K))= tilde(X)_( tilde(K))^(BS)\\ tilde(X)_( tilde(K))\partial \tilde{X}_{\tilde{K}}=\tilde{X}_{\tilde{K}}^{\mathrm{BS}} \backslash \tilde{X}_{\tilde{K}}∂X~K~=X~K~BS∖X~K~ is the boundary of the Borel-Serre compactification. The usual and compactly supported cohomology of X ~ K ~ X ~ K ~ tilde(X)_( tilde(K))\tilde{X}_{\tilde{K}}X~K~ can be simplified to some extent by applying either of the two vanishing theorems. On the other hand, the cohomology of X K X K X_(K)X_{K}XK can be shown to contribute to the cohomology of X ~ K ~ ∂ X ~ K ~ del tilde(X)_( tilde(K))\partial \tilde{X}_{\tilde{K}}∂X~K~, in some more or less controlled fashion.
Let m T m ⊂ T msubT\mathfrak{m} \subset \mathbb{T}m⊂T be a non-Eisenstein maximal ideal in the support of R Γ ( X K , Z ) R Γ X K , Z â„“ R Gamma(X_(K),Z_(â„“))R \Gamma\left(X_{K}, \mathbb{Z}_{\ell}\right)RΓ(XK,Zâ„“) and let T ( K ) m T ( K ) m T(K)_(m)\mathbb{T}(K)_{\mathfrak{m}}T(K)m denote the quotient of T T T\mathbb{T}T that acts faithfully on R Γ ( X K , Z ) m R Γ X K , Z â„“ m R Gamma(X_(K),Z_(â„“))_(m)R \Gamma\left(X_{K}, \mathbb{Z}_{\ell}\right)_{\mathfrak{m}}RΓ(XK,Zâ„“)m. In addition to the residual Galois representation ρ ¯ m ρ ¯ m bar(rho)_(m)\bar{\rho}_{\mathfrak{m}}ρ¯m, Scholze associates to m m m\mathfrak{m}m a deformation ρ m ρ m rho_(m)\rho_{\mathfrak{m}}ρm valued in T ( K ) m / I T ( K ) m / I T(K)_(m)//I\mathbb{T}(K)_{\mathfrak{m}} / IT(K)m/I, for some nilpotent ideal I I III. This was subsequently shown by Newton and Thorne in [45] to satisfy I 4 = 0 I 4 = 0 I^(4)=0I^{4}=0I4=0. In [20], we used a variant of Theorem 5.1 together with the excision sequence (6.1) to eliminate this nilpotent ideal entirely, under the assumption that â„“ â„“\ellâ„“ splits
completely in the CM field F F FFF. This leads to a more natural statement on the existence of Galois representations in this setting.
The Galois representations ρ m ρ m rho_(m)\rho_{\mathfrak{m}}ρm are expected to satisfy a certain property known as local-global compatibility, which is particularly subtle to state and prove at primes above â„“ â„“\ellâ„“. For example, after inverting â„“ â„“\ellâ„“, the ρ m ρ m rho_(m)\rho_{\mathfrak{m}}ρm are expected to be de Rham, in the sense of Fontaine, but it is less clear what the right condition should be for torsion Galois representations. In another application, Theorem 4.5 is crucially used in [1] together with the excision sequence (6.1) to prove that ρ m ρ m rho_(m)\rho_{\mathfrak{m}}ρm satisfies the expected local-global compatibility at primes above â„“ â„“\ellâ„“ in two restricted families of cases: the ordinary case and the FontaineLaffaille case. 6 6 ^(6){ }^{6}6 In joint work in progress with Newton, we should be able to extend these methods to cover significantly more.
The local-global compatibility results established in [1] are already extremely useful: they help us implement the Calegari-Geraghty method unconditionally for the first time in arbitrary dimension. A striking application is the following result.
Theorem 6.1 ([1, THEOREM 1.0.1]). Let F F FFF be a CM field and E / F E / F E//FE / FE/F be an elliptic curve that does not have complex multiplication. Then E E EEE is potentially automorphic and satisfies the Sato-Tate conjecture.
The potential automorphy of E E EEE was established at the same time in work of BoxerCalegari-Gee-Pilloni [6], who also showed the potential automorphy of abelian surfaces over totally real fields. Their work relies on the Calegari-Geraghty method for the coherent cohomology of Shimura varieties and uses a preliminary version of higher Hida theory, due to Pilloni, as a key ingredient.

ACKNOWLEDGMENTS

I am grateful to my many colleagues and collaborators who have discussed mathematics in general and Shimura varieties in particular with me - I especially want to thank Frank Calegari, Matthew Emerton, Toby Gee, Dan Gulotta, Pol van Hoften, Christian Johansson, Elena Mantovan, Sophie Morel, James Newton, Vytas Paškūnas, Vincent Pilloni, Peter Scholze, Sug Woo Shin, Matteo Tamiozzo, and Richard Taylor. I am grateful to Toby Gee, Louis Jaburi, Teruhisa Koshikawa, Peter Scholze, and Matteo Tamiozzo for their comments on an earlier draft of this article.

FUNDING

This work was partially supported by a Royal Society University Research Fellowship, by a Leverhulme Prize, and by ERC Starting Grant No. 804176.
6 Up to possibly enlarging the nilpotent ideal I I III. It is not clear how to remove the nilpotent ideal from the statement of local-global compatibility at = p â„“ = p â„“=p\ell=pâ„“=p.

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ANA CARAIANI

180 Queen's Gate, London SW7 2AZ, United Kingdom, a.caraiani @ imperial.ac.uk

ON THE BRUMER-STARK CONJECTURE AND REFINEMENTS

SAMIT DASGUPTA AND MAHESH KAKDE

Abstract

We state the Brumer-Stark conjecture and motivate it from two perspectives. Stark's perspective arose in his attempts to generalize the classical Dirichlet class number formula for the leading term of the Dedekind zeta function at s = 1 s = 1 s=1s=1s=1 (equivalently, s = 0 s = 0 s=0s=0s=0 ). Brumer's perspective arose by generalizing Stickelberger's work regarding the factorization of Gauss sums and the annihilation of class groups of cyclotomic fields. These viewpoints were synthesized by Tate, who stated the Brumer-Stark conjecture in its current form.

The conjecture considers a totally real field F F FFF and a finite abelian CM extension H / F H / F H//FH / FH/F. It states the existence of p p ppp-units in H H HHH whose valuations at places above p p ppp are related to the special values of the L L LLL-functions of the extension H / F H / F H//FH / FH/F at s = 0 s = 0 s=0s=0s=0. Essentially equivalently, the conjecture states that a Stickelberger element associated to H / F H / F H//FH / FH/F annihilates the (appropriately smoothed) class group of H H HHH.

We describe our recent proofs of the Brumer-Stark conjecture away from 2. The conjecture has been refined by many authors in multiple directions. We state some of these refinements and our results towards them. The key technique involved in the proofs is Ribet's method.

One of the refinements we discuss is an exact p p ppp-adic analytic formula for Brumer-Stark units stated by the first author and his collaborators. We describe this formula and highlight some salient points of its proof. Since the Brumer-Stark units along with other easily described elements generate the maximal abelian extension of a totally real field, our results can be viewed as an explicit class theory for such fields. This can be considered a p p ppp-adic version of Hilbert's 12th problem.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11R37; Secondary 11R42

KEYWORDS

Stark conjectures, Brumer-Stark units, Explicit class field theory

1. BACKGROUND AND MOTIVATION

Dirichlet's class number formula, conjectured for quadratic fields by Jacobi in 1832 and proven by Dirichlet in 1839, is one of the earliest examples of a relationship between leading terms of L L LLL-functions and global arithmetic invariants. Let F F FFF be a number field with ring of integers O F O F O_(F)O_{F}OF. The Dedekind zeta function associated with F F FFF is defined as
ζ F ( s ) = 0 a O F N a s , Re ( s ) > 1 ζ F ( s ) = ∑ 0 ≠ a ⊂ O F   N a − s , Re ⁡ ( s ) > 1 zeta_(F)(s)=sum_(0!=a subO_(F))Na^(-s),quad Re(s) > 1\zeta_{F}(s)=\sum_{0 \neq a \subset O_{F}} \mathrm{Na}^{-s}, \quad \operatorname{Re}(s)>1ζF(s)=∑0≠a⊂OFNa−s,Re⁡(s)>1
where a runs through the nonzero ideals in O F O F O_(F)O_{F}OF. The function ζ F ( s ) ζ F ( s ) zeta_(F)(s)\zeta_{F}(s)ζF(s) generalizes Riemann's zeta function and has a meromorphic continuation to the whole complex plane with only a simple pole at s = 1 s = 1 s=1s=1s=1. Dirichlet's class number formula, which is proved using a "geometry of numbers" approach, evaluates the residue at s = 1 s = 1 s=1s=1s=1 :
lim s 1 ( s 1 ) ζ F ( s ) = 2 r 1 ( 2 π ) r 2 R F h F w F | D F | lim s → 1   ( s − 1 ) ζ F ( s ) = 2 r 1 ( 2 Ï€ ) r 2 R F h F w F D F lim_(s rarr1)(s-1)zeta_(F)(s)=(2^(r_(1))(2pi)^(r_(2))R_(F)h_(F))/(w_(F)sqrt(|D_(F)|))\lim _{s \rightarrow 1}(s-1) \zeta_{F}(s)=\frac{2^{r_{1}}(2 \pi)^{r_{2}} R_{F} h_{F}}{w_{F} \sqrt{\left|D_{F}\right|}}lims→1(s−1)ζF(s)=2r1(2Ï€)r2RFhFwF|DF|
Here r 1 r 1 r_(1)r_{1}r1 is the number of real embeddings of F F FFF and 2 r 2 2 r 2 2r_(2)2 r_{2}2r2 is the number of complex embeddings of F F FFF. Further, h F h F h_(F)h_{F}hF and R F R F R_(F)R_{F}RF denote the class number and regulator (defined below) of F F FFF, respectively, while w F w F w_(F)w_{F}wF denotes the number of roots of unity in F F FFF and D F D F D_(F)D_{F}DF is the discriminant of F / Q F / Q F//QF / \mathbf{Q}F/Q. The meromorphic function ζ F ( s ) ζ F ( s ) zeta_(F)(s)\zeta_{F}(s)ζF(s) satisfies a functional equation relating ζ F ( s ) ζ F ( s ) zeta_(F)(s)\zeta_{F}(s)ζF(s) and ζ F ( 1 s ) ζ F ( 1 − s ) zeta_(F)(1-s)\zeta_{F}(1-s)ζF(1−s). Using this functional equation, Dirichlet's class number formula can be restated as giving the leading term of the Taylor expansion of ζ F ( s ) ζ F ( s ) zeta_(F)(s)\zeta_{F}(s)ζF(s) at s = 0 s = 0 s=0s=0s=0 :
(1.1) ζ F ( s ) = h F R F w F s r 1 + r 2 1 + O ( s r 1 + r 2 ) (1.1) ζ F ( s ) = − h F R F w F s r 1 + r 2 − 1 + O s r 1 + r 2 {:(1.1)zeta_(F)(s)=-(h_(F)R_(F))/(w_(F))s^(r_(1)+r_(2)-1)+O(s^(r_(1)+r_(2))):}\begin{equation*} \zeta_{F}(s)=-\frac{h_{F} R_{F}}{w_{F}} s^{r_{1}+r_{2}-1}+O\left(s^{r_{1}+r_{2}}\right) \tag{1.1} \end{equation*}(1.1)ζF(s)=−hFRFwFsr1+r2−1+O(sr1+r2)
Artin described a theory of L L LLL-functions generalizing the Dedekind zeta function. Let G F G F G_(F)G_{F}GF be the absolute Galois group of F F FFF. A Dirichlet character for F F FFF (or a degree 1 Artin character of F F FFF ) is a homomorphism χ : G F C × Ï‡ : G F → C × chi:G_(F)rarrC^(xx)\chi: G_{F} \rightarrow \mathbf{C}^{\times}χ:GF→C×with finite image. Class field theory identifies χ χ chi\chiχ with a function, again denoted by χ χ chi\chiχ, from the set of nonzero ideals of O F O F O_(F)O_{F}OF to C C C\mathbf{C}C. Define
L ( χ , s ) = 0 a O F χ ( a ) N a s , Re ( s ) > 1 L ( χ , s ) = ∑ 0 ≠ a ⊂ O F   χ ( a ) N a − s , Re ⁡ ( s ) > 1 L(chi,s)=sum_(0!=a subO_(F))chi(a)Na^(-s),quad Re(s) > 1L(\chi, s)=\sum_{0 \neq a \subset O_{F}} \chi(a) \mathrm{Na}^{-s}, \quad \operatorname{Re}(s)>1L(χ,s)=∑0≠a⊂OFχ(a)Na−s,Re⁡(s)>1
Again L ( χ , s ) L ( χ , s ) L(chi,s)L(\chi, s)L(χ,s) has a meromorphic continuation to the whole complex plane with only a simple pole at s = 1 s = 1 s=1s=1s=1 if χ χ chi\chiχ is trivial. If H / F H / F H//FH / FH/F is a Galois extension with finite abelian Galois group G = Gal ( H / F ) G = Gal ⁡ ( H / F ) G=Gal(H//F)G=\operatorname{Gal}(H / F)G=Gal⁡(H/F), then we can view any character χ G ^ = Hom ( G , C ) χ ∈ G ^ = Hom ⁡ G , C ∗ chi in hat(G)=Hom(G,C^(**))\chi \in \hat{G}=\operatorname{Hom}\left(G, \mathbf{C}^{*}\right)χ∈G^=Hom⁡(G,C∗) as a Dirichlet character for F F FFF, and we have the Artin factorization formula
(1.2) ζ H ( s ) = χ G ^ L ( χ , s ) (1.2) ζ H ( s ) = ∏ χ ∈ G ^   L ( χ , s ) {:(1.2)zeta_(H)(s)=prod_(chi in hat(G))L(chi","s):}\begin{equation*} \zeta_{H}(s)=\prod_{\chi \in \hat{G}} L(\chi, s) \tag{1.2} \end{equation*}(1.2)ζH(s)=∏χ∈G^L(χ,s)
Dirichlet's class number formula (1.1) for the field H H HHH gives the leading term of the left-hand side of (1.2) at s = 0 s = 0 s=0s=0s=0. Stark asked for an analogous formula for L ( χ , s ) L ( χ , s ) L(chi,s)L(\chi, s)L(χ,s) at s = 0 s = 0 s=0s=0s=0 for each character χ χ chi\chiχ, thereby giving a canonical factorization of the term h H R H / w H h H R H / w H h_(H)R_(H)//w_(H)h_{H} R_{H} / w_{H}hHRH/wH. This led to the formulation of the abelian Stark conjecture, which we state in Section 2. This statement involves the choice of places of F F FFF that split completely in H H HHH. After stating Stark's conjecture, we restrict in the remainder of the paper to the case that the splitting places of F F FFF are finite.
Since the associated L L LLL-values here are algebraic, one can make progress on the conjectures through p p ppp-adic techniques such as p p ppp-adic Galois cohomology. To obtain nonzero L L LLL-values (and hence have nontrivial statements), parity conditions force us to restrict to the setting that F F FFF is a totally real field and H H HHH is a CM field.
Stark's conjecture at finite places has a natural restatement in terms of annihilators of class groups as formulated in the Brumer-Stark conjecture. We recall the statement and its refinements in Sections 4-5. The rest of the paper is taken up in describing the statement and proofs of our results toward the Brumer-Stark conjecture and its refinements.

2. STARK'S CONJECTURE

Let us first reformulate Dirichlet's class number formula.
For any place w w www of F F FFF we normalize the absolute value | | w : F w R | ⋅ | w : F w ∗ → R |*|_(w):F_(w)^(**)rarrR|\cdot|_{w}: F_{w}^{*} \rightarrow \mathbf{R}|⋅|w:Fw∗→R by
| u | w = { | u | if w is real | u | 2 if w is complex N w ord w ( u ) if w is nonarchimedean. | u | w = | u |  if  w  is real  | u | 2  if  w  is complex  N w − ord w ⁡ ( u )  if  w  is nonarchimedean.  |u|_(w)={[|u|," if "w" is real "],[|u|^(2)," if "w" is complex "],[Nw^(-ord_(w)(u))," if "w" is nonarchimedean. "]:}|u|_{w}= \begin{cases}|u| & \text { if } w \text { is real } \\ |u|^{2} & \text { if } w \text { is complex } \\ \mathrm{N} w^{-\operatorname{ord}_{w}(u)} & \text { if } w \text { is nonarchimedean. }\end{cases}|u|w={|u| if w is real |u|2 if w is complex Nw−ordw⁡(u) if w is nonarchimedean. 
For a finite set of places S S SSS of F F FFF, let X S X S X_(S)X_{S}XS denote the degree zero subgroup of the free abelian group on S S SSS. Let u 1 , , u r 1 + r 2 1 u 1 , … , u r 1 + r 2 − 1 u_(1),dots,u_(r_(1)+r_(2)-1)u_{1}, \ldots, u_{r_{1}+r_{2}-1}u1,…,ur1+r2−1 be a set of generators of the free abelian group O F / μ F O F ∗ / μ F O_(F)^(**)//mu_(F)O_{F}^{*} / \mu_{F}OF∗/μF. Let S S ∞ S_(oo)S_{\infty}S∞ be the set of archimedean places of F F FFF.
The Dirichlet regulator map
O F / μ F R X S , u w S log | u | w w O F ∗ / μ F → R X S ∞ , u ↦ ∑ w ∈ S ∞   log ⁡ | u | w â‹… w O_(F)^(**)//mu_(F)rarrRX_(S_(oo)),quad u|->sum_(w inS_(oo))log |u|_(w)*wO_{F}^{*} / \mu_{F} \rightarrow \mathbf{R} X_{S_{\infty}}, \quad u \mapsto \sum_{w \in S_{\infty}} \log |u|_{w} \cdot wOF∗/μF→RXS∞,u↦∑w∈S∞log⁡|u|wâ‹…w
induces an isomorphism R O F R X S R O F ∗ → R X S ∞ RO_(F)^(**)rarrRX_(S_(oo))\mathbf{R} O_{F}^{*} \rightarrow \mathbf{R} X_{S_{\infty}}ROF∗→RXS∞. Here and throughout, R X S R X S ∞ RX_(S_(oo))\mathbf{R} X_{S_{\infty}}RXS∞ denotes R Z X S R ⊗ Z X S ∞ Rox_(Z)X_(S_(oo))\mathbf{R} \otimes_{\mathbf{Z}} X_{S_{\infty}}R⊗ZXS∞, etc. Let w 1 , , w r 1 + r 2 w 1 , … , w r 1 + r 2 w_(1),dots,w_(r_(1)+r_(2))w_{1}, \ldots, w_{r_{1}+r_{2}}w1,…,wr1+r2 denote the archimedean places of F F FFF. Then
(2.1) { w i w 1 : 2 i r + 1 } , r = r 1 + r 2 1 (2.1) w i − w 1 : 2 ≤ i ≤ r + 1 , r = r 1 + r 2 − 1 {:(2.1){w_(i)-w_(1):2 <= i <= r+1}","quad r=r_(1)+r_(2)-1:}\begin{equation*} \left\{w_{i}-w_{1}: 2 \leq i \leq r+1\right\}, \quad r=r_{1}+r_{2}-1 \tag{2.1} \end{equation*}(2.1){wi−w1:2≤i≤r+1},r=r1+r2−1
is an integral basis of X S X S ∞ X_(S_(oo))X_{S_{\infty}}XS∞. Let R F R F R_(F)R_{F}RF be the absolute value of the determinant of the isomorphism between R O F R O F ∗ RO_(F)^(**)\mathbf{R} O_{F}^{*}ROF∗ and R X S R X S ∞ RX_(S_(oo))\mathbf{R} X_{S_{\infty}}RXS∞ with respect to the bases { u 1 , , u r 1 + r 2 1 } u 1 , … , u r 1 + r 2 − 1 {u_(1),dots,u_(r_(1)+r_(2)-1)}\left\{u_{1}, \ldots, u_{r_{1}+r_{2}-1}\right\}{u1,…,ur1+r2−1} and (2.1), respectively. Up to a sign, Dirichlet's class number formula can be restated as follows:
(i) The rational structure Q O F Q O F ∗ QO_(F)^(**)\mathbf{Q} O_{F}^{*}QOF∗ on the left-hand side corresponds to the structure ζ F ( r ) ( 0 ) Q X S ζ F ( r ) ( 0 ) Q X S ∞ zeta_(F)^((r))(0)QX_(S_(oo))\zeta_{F}^{(r)}(0) \mathbf{Q} X_{S_{\infty}}ζF(r)(0)QXS∞ on the right-hand side.
(ii) The integral structure O F / μ F O F ∗ / μ F O_(F)^(**)//mu_(F)O_{F}^{*} / \mu_{F}OF∗/μF on the left-hand side corresponds to the structure ζ F ( r ) ( 0 ) X S ζ F ( r ) ( 0 ) X S ∞ zeta_(F)^((r))(0)X_(S_(oo))\zeta_{F}^{(r)}(0) X_{S_{\infty}}ζF(r)(0)XS∞ on the right-hand side.
Motivated by this reformulation, we present Stark's conjecture and its integral refinement due to Rubin. For details, see [41]. Let F F FFF be a number field of degree n n nnn and let H / F H / F H//FH / FH/F be a finite Galois extension with G = Gal ( H / F ) G = Gal ⁡ ( H / F ) G=Gal(H//F)G=\operatorname{Gal}(H / F)G=Gal⁡(H/F) abelian. Let S , T S , T S,TS, TS,T be two finite disjoint sets of places of F F FFF satisfying the following conditions:
(1) S S SSS contains the sets S S ∞ S_(oo)S_{\infty}S∞ of archimedean places and S ram S ram  S_("ram ")S_{\text {ram }}Sram  of places ramified in H H HHH.
(2) T T TTT contains at least two primes of different residue characteristic or at least one prime of residue characteristic larger than n + 1 n + 1 n+1n+1n+1, where n = [ F : Q ] n = [ F : Q ] n=[F:Q]n=[F: \mathbf{Q}]n=[F:Q].
For any character χ G ^ = Hom ( G , C ) χ ∈ G ^ = Hom ⁡ G , C ∗ chi in hat(G)=Hom(G,C^(**))\chi \in \hat{G}=\operatorname{Hom}\left(G, \mathbf{C}^{*}\right)χ∈G^=Hom⁡(G,C∗), define the S S SSS-depleted, T T TTT-smoothed L L LLL-function
L S , T ( χ , s ) = p S 1 1 χ ( p ) N p s p T ( 1 χ ( p ) N p 1 s ) , Re ( s ) > 1 L S , T ( χ , s ) = ∏ p ∉ S   1 1 − χ ( p ) N p − s ∏ p ∈ T   1 − χ ( p ) N p 1 − s , Re ⁡ ( s ) > 1 L_(S,T)(chi,s)=prod_(p!in S)(1)/(1-chi(p)Np^(-s))prod_(pin T)(1-chi(p)Np^(1-s)),quad Re(s) > 1L_{S, T}(\chi, s)=\prod_{\mathfrak{p} \notin S} \frac{1}{1-\chi(\mathfrak{p}) \mathrm{Np}^{-s}} \prod_{\mathfrak{p} \in T}\left(1-\chi(\mathfrak{p}) \mathrm{Np}^{1-s}\right), \quad \operatorname{Re}(s)>1LS,T(χ,s)=∏p∉S11−χ(p)Np−s∏p∈T(1−χ(p)Np1−s),Re⁡(s)>1
The function L S , T ( χ , s ) L S , T ( χ , s ) L_(S,T)(chi,s)L_{S, T}(\chi, s)LS,T(χ,s) extends by analytic continuation to a holomorphic function on C C C\mathbf{C}C. The Stickelberger element associated to this data is the unique group-ring element Θ S , T ( H / F , s ) C [ G ] Θ S , T ( H / F , s ) ∈ C [ G ] Theta_(S,T)(H//F,s)inC[G]\Theta_{S, T}(H / F, s) \in \mathbf{C}[G]ΘS,T(H/F,s)∈C[G] satisfying
χ ( Θ S , T ( H / F , s ) ) = L S , T ( χ 1 , s ) for all χ G ^ χ Θ S , T ( H / F , s ) = L S , T χ − 1 , s  for all  χ ∈ G ^ chi(Theta_(S,T)(H//F,s))=L_(S,T)(chi^(-1),s)quad" for all "chi in hat(G)\chi\left(\Theta_{S, T}(H / F, s)\right)=L_{S, T}\left(\chi^{-1}, s\right) \quad \text { for all } \chi \in \hat{G}χ(ΘS,T(H/F,s))=LS,T(χ−1,s) for all χ∈G^
Let S H S H S_(H)S_{H}SH denote the set of places of H H HHH above those in S S SSS, and similarly for T H T H T_(H)T_{H}TH. Define
U S , T = { x H : ord w ( x ) 0 for all w S H and x 1 ( mod T H ) } U S , T = x ∈ H ∗ : ord w ⁡ ( x ) ≥ 0  for all  w ∉ S H  and  x ≡ 1 mod T H U_(S,T)={x inH^(**):ord_(w)(x) >= 0" for all "w!inS_(H)" and "x-=1(modT_(H))}U_{S, T}=\left\{x \in H^{*}: \operatorname{ord}_{w}(x) \geq 0 \text { for all } w \notin S_{H} \text { and } x \equiv 1\left(\bmod T_{H}\right)\right\}US,T={x∈H∗:ordw⁡(x)≥0 for all w∉SH and x≡1(modTH)}
The condition on T T TTT ensures that U S , T U S , T U_(S,T)U_{S, T}US,T does not have any torsion. The Galois equivariant version of Dirichlet's unit theorem gives an R [ G ] R [ G ] R[G]\mathbf{R}[G]R[G]-module isomorphism
λ : R U S , T R X S H (2.2) λ ( u ) = w S H log ( | u | w ) w λ : R U S , T → R X S H (2.2) λ ( u ) = ∑ w ∈ S H   log ⁡ | u | w â‹… w {:[lambda:RU_(S,T)rarrRX_(S_(H))],[(2.2)lambda(u)=sum_(w inS_(H))log(|u|_(w))*w]:}\begin{align*} & \lambda: \mathbf{R} U_{S, T} \rightarrow \mathbf{R} X_{S_{H}} \\ & \lambda(u)=\sum_{w \in S_{H}} \log \left(|u|_{w}\right) \cdot w \tag{2.2} \end{align*}λ:RUS,T→RXSH(2.2)λ(u)=∑w∈SHlog⁡(|u|w)â‹…w
Suppose that exactly r r rrr places v 1 , , v r S v 1 , … , v r ∈ S v_(1),dots,v_(r)in Sv_{1}, \ldots, v_{r} \in Sv1,…,vr∈S split completely in H H HHH and # S r + 1 # S ≥ r + 1 #S >= r+1\# S \geq r+1#S≥r+1. The order of vanishing of L S , T ( χ , s ) L S , T ( χ , s ) L_(S,T)(chi,s)L_{S, T}(\chi, s)LS,T(χ,s) at s = 0 s = 0 s=0s=0s=0 is given by
r ( χ ) = dim C ( C U S , T ) ( χ ) = { # { v S : χ ( v ) = 1 } if χ 1 # S 1 if χ = 1 r ( χ ) = dim C ⁡ C U S , T ( χ ) = # { v ∈ S : χ ( v ) = 1 }  if  χ ≠ 1 # S − 1  if  χ = 1 r(chi)=dim_(C)(CU_(S,T))^((chi))={[#{v in S:chi(v)=1}," if "chi!=1],[#S-1," if "chi=1]:}r(\chi)=\operatorname{dim}_{\mathbf{C}}\left(\mathbf{C} U_{S, T}\right)^{(\chi)}= \begin{cases}\#\{v \in S: \chi(v)=1\} & \text { if } \chi \neq 1 \\ \# S-1 & \text { if } \chi=1\end{cases}r(χ)=dimC⁡(CUS,T)(χ)={#{v∈S:χ(v)=1} if χ≠1#S−1 if χ=1
whence r ( χ ) r r ( χ ) ≥ r r(chi) >= rr(\chi) \geq rr(χ)≥r for all χ G ^ χ ∈ G ^ chi in hat(G)\chi \in \hat{G}χ∈G^. Stark's conjecture predicts that the r r rrr th derivative Θ S , T ( r ) ( H / F , 0 ) Θ S , T ( r ) ( H / F , 0 ) Theta_(S,T)^((r))(H//F,0)\Theta_{S, T}^{(r)}(H / F, 0)ΘS,T(r)(H/F,0) captures the "non-rationality" of the map λ λ lambda\lambdaλ.
Conjecture 2.1 (Stark). We have
Θ S , T ( r ) ( H / F , 0 ) Q r X S H λ ( Q r U S , T ) Θ S , T ( r ) ( H / F , 0 ) â‹… Q â‹€ r   X S H ⊂ λ Q â‹€ r   U S , T Theta_(S,T)^((r))(H//F,0)*Q^^^rX_(S_(H))sub lambda(Q^^^rU_(S,T))\Theta_{S, T}^{(r)}(H / F, 0) \cdot \mathbf{Q} \bigwedge^{r} X_{S_{H}} \subset \lambda\left(\mathbf{Q} \bigwedge^{r} U_{S, T}\right)ΘS,T(r)(H/F,0)â‹…Qâ‹€rXSH⊂λ(Qâ‹€rUS,T)
Concretely, this states that for each character χ χ chi\chiχ of G G GGG with r ( χ ) = r r ( χ ) = r r(chi)=rr(\chi)=rr(χ)=r, the nonzero number L S , T ( r ) ( χ 1 , s ) L S , T ( r ) χ − 1 , s L_(S,T)^((r))(chi^(-1),s)L_{S, T}^{(r)}\left(\chi^{-1}, s\right)LS,T(r)(χ−1,s) lies in the one-dimensional Q Q Q\mathbf{Q}Q-vector space spanned by λ ( r ( U S , T ( χ ) ) ) λ â‹€ r   U S , T ( χ ) lambda(^^^r(U_(S,T)^((chi))))\lambda\left(\bigwedge^{r}\left(U_{S, T}^{(\chi)}\right)\right)λ(â‹€r(US,T(χ))).
Let us reformulate Conjecture 2.1 in terms of the existence of special elements. Write X S H = Hom ( X S H , Z [ G ] ) X S H ∗ = Hom ⁡ X S H , Z [ G ] X_(S_(H))^(**)=Hom(X_(S_(H)),Z[G])X_{S_{H}}^{*}=\operatorname{Hom}\left(X_{S_{H}}, \mathbb{Z}[G]\right)XSH∗=Hom⁡(XSH,Z[G]). For φ r X S H φ ∈ ∧ r X S H ∗ varphi in^^^(r)X_(S_(H))^(**)\varphi \in \wedge^{r} X_{S_{H}}^{*}φ∈∧rXSH∗, there is a determinant map
r X S H × r X S H Z [ G ] â‹€ r   X S H × â‹€ r   X S H ∗ → Z [ G ] ^^^rX_(S_(H))xx^^^rX_(S_(H))^(**)rarrZ[G]\bigwedge^{r} X_{S_{H}} \times \bigwedge^{r} X_{S_{H}}^{*} \rightarrow \mathbb{Z}[G]â‹€rXSH×⋀rXSH∗→Z[G]
defined by
( x 1 x r , φ 1 φ r ) φ 1 φ r ( x 1 x r ) = det ( φ i ( x j ) ) i , j x 1 ∧ ⋯ ∧ x r , φ 1 ∧ ⋯ ∧ φ r ↦ φ 1 ∧ ⋯ ∧ φ r x 1 ∧ ⋯ ∧ x r = det ⁡ φ i x j i , j (x_(1)^^cdots^^x_(r),varphi_(1)^^cdots^^varphi_(r))|->varphi_(1)^^cdots^^varphi_(r)(x_(1)^^cdots^^x_(r))=det (varphi_(i)(x_(j)))_(i,j)\left(x_{1} \wedge \cdots \wedge x_{r}, \varphi_{1} \wedge \cdots \wedge \varphi_{r}\right) \mapsto \varphi_{1} \wedge \cdots \wedge \varphi_{r}\left(x_{1} \wedge \cdots \wedge x_{r}\right)=\operatorname{det}\left(\varphi_{i}\left(x_{j}\right)\right)_{i, j}(x1∧⋯∧xr,φ1∧⋯∧φr)↦φ1∧⋯∧φr(x1∧⋯∧xr)=det⁡(φi(xj))i,j
We extend the determinant map to R R R\mathbf{R}R-linearizations. We fix a place w i w i w_(i)w_{i}wi of H H HHH above each v i v i v_(i)v_{i}vi. Let w i X S H w i ∗ ∈ X S H ∗ w_(i)^(**)inX_(S_(H))^(**)w_{i}^{*} \in X_{S_{H}}^{*}wi∗∈XSH∗ be induced by
w i ( w ) = γ G : γ w i = w γ w i ∗ ( w ) = ∑ γ ∈ G : γ w i = w   γ w_(i)^(**)(w)=sum_(gamma in G:gammaw_(i)=w)gammaw_{i}^{*}(w)=\sum_{\gamma \in G: \gamma w_{i}=w} \gammawi∗(w)=∑γ∈G:γwi=wγ
Conjecture 2.2 (Stark). Put φ = w 1 w r φ = w 1 ∗ ∧ ⋯ ∧ w r ∗ varphi=w_(1)^(**)^^cdots^^w_(r)^(**)\varphi=w_{1}^{*} \wedge \cdots \wedge w_{r}^{*}φ=w1∗∧⋯∧wr∗. There exists u Q r U S , T u ∈ Q ∧ r U S , T u inQ^^^(r)U_(S,T)u \in \mathbf{Q} \wedge^{r} U_{S, T}u∈Q∧rUS,T such that
φ ( λ ( u ) ) = Θ S , T ( r ) ( H / F , 0 ) φ ( λ ( u ) ) = Θ S , T ( r ) ( H / F , 0 ) varphi(lambda(u))=Theta_(S,T)^((r))(H//F,0)\varphi(\lambda(u))=\Theta_{S, T}^{(r)}(H / F, 0)φ(λ(u))=ΘS,T(r)(H/F,0)
The equivalence of Conjectures 2.1 and 2.2 is proven in [41, PROPOSITION 2.4].
We are now ready to state the integral version of Stark's conjecture. In the rank r = 1 r = 1 r=1r=1r=1 case, Stark proposed the statement that u u uuu in Conjecture 2.2 lies in U S , T U S , T U_(S,T)U_{S, T}US,T. This is the famous "rank 1 abelian Stark conjecture." In the higher rank case, the obvious generalization u r U S , T u ∈ ⋀ r   U S , T u in^^^rU_(S,T)u \in \bigwedge^{r} U_{S, T}u∈⋀rUS,T is not true, as was experimentally observed by Rubin [41]. Rubin defined a lattice, nowadays called "Rubin's lattice" and conjectured that it contains the element u u uuu.
Put U S , T = Hom Z [ G ] ( U S , T , Z [ G ] ) U S , T ∗ = Hom Z [ G ] ⁡ U S , T , Z [ G ] U_(S,T)^(**)=Hom_(Z[G])(U_(S,T),Z[G])U_{S, T}^{*}=\operatorname{Hom}_{\mathbf{Z}[G]}\left(U_{S, T}, \mathbf{Z}[G]\right)US,T∗=HomZ[G]⁡(US,T,Z[G]).
The r r rrr th exterior bidual of U S , T U S , T U_(S,T)U_{S, T}US,T (see [7] for a more general study and the initiation of this terminology) is defined by
r U S , T = ( r U S , T ) { x r Q U S , T : φ ( x ) Z [ G ] for all φ r U S , T } â‹‚ r   U S , T = â‹€ r   U S , T ∗ ∗ ≅ x ∈ â‹€ r   Q U S , T : φ ( x ) ∈ Z [ G ]  for all  φ ∈ â‹€ r   U S , T ∗ nnnrU_(S,T)=(^^^rU_(S,T)^(**))^(**)~={x in^^^rQU_(S,T):varphi(x)inZ[G]" for all "varphi in^^^rU_(S,T)^(**)}\bigcap^{r} U_{S, T}=\left(\bigwedge^{r} U_{S, T}^{*}\right)^{*} \cong\left\{x \in \bigwedge^{r} \mathbf{Q} U_{S, T}: \varphi(x) \in \mathbf{Z}[G] \text { for all } \varphi \in \bigwedge^{r} U_{S, T}^{*}\right\}â‹‚rUS,T=(â‹€rUS,T∗)∗≅{x∈⋀rQUS,T:φ(x)∈Z[G] for all φ∈⋀rUS,T∗}
We would like to consider only the "rank r r rrr " component of this bidual. To this end, for each character χ G ^ χ ∈ G ^ chi in hat(G)\chi \in \hat{G}χ∈G^ consider the associated idempotent
e χ = 1 # G g G χ ( g ) g 1 C [ G ] e χ = 1 # G ∑ g ∈ G   χ ( g ) g − 1 ∈ C [ G ] e_(chi)=(1)/(#G)sum_(g in G)chi(g)g^(-1)inC[G]e_{\chi}=\frac{1}{\# G} \sum_{g \in G} \chi(g) g^{-1} \in \mathbf{C}[G]eχ=1#G∑g∈Gχ(g)g−1∈C[G]
Define e r = e χ Q [ G ] e r = ∑ e χ ∈ Q [ G ] e_(r)=sume_(chi)inQ[G]e_{r}=\sum e_{\chi} \in \mathbf{Q}[G]er=∑eχ∈Q[G], where the sum extends over the set
{ χ G ^ : L S , T ( r ) ( χ , 0 ) 0 } = { χ G ^ : χ ( G v ) 1 , v S { v 1 , , v r } } χ ∈ G ^ : L S , T ( r ) ( χ , 0 ) ≠ 0 = χ ∈ G ^ : χ G v ≠ 1 , v ∈ S ∖ v 1 , … , v r {chi in( hat(G)):L_(S,T)^((r))(chi,0)!=0}={chi in( hat(G)):chi(G_(v))!=1,v in S\\{v_(1),dots,v_(r)}}\left\{\chi \in \hat{G}: L_{S, T}^{(r)}(\chi, 0) \neq 0\right\}=\left\{\chi \in \hat{G}: \chi\left(G_{v}\right) \neq 1, v \in S \backslash\left\{v_{1}, \ldots, v_{r}\right\}\right\}{χ∈G^:LS,T(r)(χ,0)≠0}={χ∈G^:χ(Gv)≠1,v∈S∖{v1,…,vr}}
Define Rubin's lattice by
L ( r ) U S , T = ( r U S , T ) e r ( Q r U S , T ) L ( r ) U S , T = ⋂ r   U S , T ∩ e r Q ⋀ r   U S , T ∗ L^((r))U_(S,T)=(nnnrU_(S,T))nne_(r)(Q^^^rU_(S,T)^(**))\mathscr{L}^{(r)} U_{S, T}=\left(\bigcap^{r} U_{S, T}\right) \cap e_{r}\left(\mathbf{Q} \bigwedge^{r} U_{S, T}^{*}\right)L(r)US,T=(⋂rUS,T)∩er(Q⋀rUS,T∗)
The following is Rubin's higher rank integral Stark conjecture.
Conjecture 2.3 ([41], Conjecture B'). Put φ = w 1 w r φ = w 1 ∗ ∧ ⋯ ∧ w r ∗ varphi=w_(1)^(**)^^cdots^^w_(r)^(**)\varphi=w_{1}^{*} \wedge \cdots \wedge w_{r}^{*}φ=w1∗∧⋯∧wr∗. There exists u L ( r ) U S , T u ∈ L ( r ) U S , T u inL^((r))U_(S,T)u \in \mathscr{L}^{(r)} U_{S, T}u∈L(r)US,T such that
φ ( λ ( u ) ) = Θ S , T ( r ) ( H / F , 0 ) φ ( λ ( u ) ) = Θ S , T ( r ) ( H / F , 0 ) varphi(lambda(u))=Theta_(S,T)^((r))(H//F,0)\varphi(\lambda(u))=\Theta_{S, T}^{(r)}(H / F, 0)φ(λ(u))=ΘS,T(r)(H/F,0)

3. STARK'S CONJECTURES AT FINITE PLACES

We now assume that the totally split places v 1 , , v r v 1 , … , v r v_(1),dots,v_(r)v_{1}, \ldots, v_{r}v1,…,vr from the previous section are all finite. This happens only when F F FFF is a totally real field and H H HHH is totally complex. In fact, the fixed fields of characters with nonvanishing L L LLL-functions at 0 are C M C M CM\mathrm{CM}CM fields, so we restrict to the setting where F F FFF is totally real and H H HHH is C M C M CM\mathrm{CM}CM for the remainder of the article. We also
enact a slight notational change and write the set denoted S S SSS in the previous sections as S S ′ S^(')S^{\prime}S′, and let S = S { v 1 , , v r } S = S ′ ∖ v 1 , … , v r S=S^(')\\{v_(1),dots,v_(r)}S=S^{\prime} \backslash\left\{v_{1}, \ldots, v_{r}\right\}S=S′∖{v1,…,vr}. The reason for this is that we now still have S S S ram S ⊃ S ∞ ∪ S ram  S supS_(oo)uuS_("ram ")S \supset S_{\infty} \cup S_{\text {ram }}S⊃S∞∪Sram .
As we explain, in this setting Conjecture 2.2 for S S ′ S^(')S^{\prime}S′ follows from a classical rationality result of Klingen-Siegel, though the integral refinement in Conjecture 2.3 remains a nontrivial statement. For a fixed place w w www of H H HHH, we have
(3.1) log | u | w = ord w ( u ) log N w (3.1) log ⁡ | u | w = − ord w ⁡ ( u ) log ⁡ N w {:(3.1)log |u|_(w)=-ord_(w)(u)log Nw:}\begin{equation*} \log |u|_{w}=-\operatorname{ord}_{w}(u) \log \mathrm{N} w \tag{3.1} \end{equation*}(3.1)log⁡|u|w=−ordw⁡(u)log⁡Nw
Since the Euler factors at the v i v i v_(i)v_{i}vi are equal to ( 1 N v i s ) = ( 1 N w i s ) 1 − N v i − s = 1 − N w i − s (1-Nv_(i)^(-s))=(1-Nw_(i)^(-s))\left(1-\mathrm{N} v_{i}^{-s}\right)=\left(1-\mathrm{N} w_{i}^{-s}\right)(1−Nvi−s)=(1−Nwi−s), we also have
(3.2) Θ S , T ( r ) ( H / F , 0 ) = Θ S , T ( H / F , 0 ) i = 1 r log N w i (3.2) Θ S ′ , T ( r ) ( H / F , 0 ) = Θ S , T ( H / F , 0 ) â‹… ∏ i = 1 r   log ⁡ N w i {:(3.2)Theta_(S^('),T)^((r))(H//F","0)=Theta_(S,T)(H//F","0)*prod_(i=1)^(r)log Nw_(i):}\begin{equation*} \Theta_{S^{\prime}, T}^{(r)}(H / F, 0)=\Theta_{S, T}(H / F, 0) \cdot \prod_{i=1}^{r} \log \mathrm{N} w_{i} \tag{3.2} \end{equation*}(3.2)ΘS′,T(r)(H/F,0)=ΘS,T(H/F,0)⋅∏i=1rlog⁡Nwi
Theorem 3.1 (Klingen-Siegel). We have Θ S , T := Θ S , T ( H / F , 0 ) Q [ G ] Θ S , T := Θ S , T ( H / F , 0 ) ∈ Q [ G ] Theta_(S,T):=Theta_(S,T)(H//F,0)inQ[G]\Theta_{S, T}:=\Theta_{S, T}(H / F, 0) \in \mathbf{Q}[G]ΘS,T:=ΘS,T(H/F,0)∈Q[G].
With e r e r e_(r)e_{r}er as in the previous section, we are then led to define a map over Q Q Q\mathbf{Q}Q
λ Q : e r ( Q U S , T ) e r ( Q X S H ) , λ Q ( u ) = w v i some i ord w ( u ) w λ Q : e r Q U S ′ , T → e r Q X S H ′ , λ Q ( u ) = ∑ w ∣ v i  some  i   ord w ⁡ ( u ) â‹… w lambda_(Q):e_(r)(QU_(S^('),T))rarre_(r)(QX_(S_(H)^('))),quadlambda_(Q)(u)=sum_(w∣v_(i)" some "i)ord_(w)(u)*w\lambda_{\mathbf{Q}}: e_{r}\left(\mathbf{Q} U_{S^{\prime}, T}\right) \rightarrow e_{r}\left(\mathbf{Q} X_{S_{H}^{\prime}}\right), \quad \lambda_{\mathbf{Q}}(u)=\sum_{w \mid v_{i} \text { some } i} \operatorname{ord}_{w}(u) \cdot wλQ:er(QUS′,T)→er(QXSH′),λQ(u)=∑w∣vi some iordw⁡(u)â‹…w
Note that e r ( Q X S H ) e r Q X S H ′ e_(r)(QX_(S_(H)^(')))e_{r}\left(\mathbf{Q} X_{S_{H}^{\prime}}\right)er(QXSH′) is the Q Q Q\mathbf{Q}Q-vector space generated by the places of H H HHH above the v i v i v_(i)v_{i}vi. The map λ Q λ Q lambda_(Q)\lambda_{\mathbf{Q}}λQ is a Q [ G ] Q [ G ] Q[G]\mathbf{Q}[G]Q[G]-module isomorphism, and it induces an isomorphism on the free rank one Q [ G ] Q [ G ] Q[G]\mathbf{Q}[G]Q[G]-modules obtained by taking r r rrr th wedge powers. In view of (3.1), the map on r r rrr th wedge powers induced by the map λ λ lambda\lambdaλ of (2.2), when restricted to e r ( Q r U S , T ) e r Q ∧ r U S ′ , T e_(r)(Q^^^(r)U_(S^('),T))e_{r}\left(\mathbf{Q} \wedge^{r} U_{S^{\prime}, T}\right)er(Q∧rUS′,T), is equal to ( i = 1 r log N w i ) λ Q ∏ i = 1 r   log ⁡ N w i â‹… λ Q (prod_(i=1)^(r)log Nw_(i))*lambda_(Q)\left(\prod_{i=1}^{r} \log \mathrm{N} w_{i}\right) \cdot \lambda_{\mathbf{Q}}(∏i=1rlog⁡Nwi)⋅λQ. Conjecture 2.2 follows from this observation together with (3.2), since Theorem 3.1 implies the existence of u e r ( Q r U S , T ) u ∈ e r Q â‹€ r   U S ′ , T u ine_(r)(Q^^^rU_(S^('),T))u \in e_{r}\left(\mathbf{Q} \bigwedge^{r} U_{S^{\prime}, T}\right)u∈er(Qâ‹€rUS′,T) such that
φ ( λ Q ( u ) ) = Θ S , T φ λ Q ( u ) = Θ S , T varphi(lambda_(Q)(u))=Theta_(S,T)\varphi\left(\lambda_{\mathbf{Q}}(u)\right)=\Theta_{S, T}φ(λQ(u))=ΘS,T
Here φ = w 1 w r φ = w 1 ∗ ∧ ⋯ ∧ w r ∗ varphi=w_(1)^(**)^^cdots^^w_(r)^(**)\varphi=w_{1}^{*} \wedge \cdots \wedge w_{r}^{*}φ=w1∗∧⋯∧wr∗ as in the statement of the conjecture.
On the other hand, the integral statement in Conjecture 2.3 lies deeper. We first note the following celebrated theorem of Deligne-Ribet [21] and Cassou-Noguès [8] refining the Klingen-Siegel theorem. The condition on the set T T TTT stated in Section 2 is crucial in this result. We remark that Deligne-RIbet prove their result using Hilbert modular forms, as an integral refinement of the strategy of the strategy established earlier by Siegel. This theme reappears in our own work described in § 6 § 6 §6\S 6§6.
Theorem 3.2. We have Θ S , T Z [ G ] Θ S , T ∈ Z [ G ] Theta_(S,T)inZ[G]\Theta_{S, T} \in \mathbf{Z}[G]ΘS,T∈Z[G].
Conjecture 2.3 in this setting is known as the Rubin-Brumer-Stark conjecture:
Conjecture 3.3 (Rubin-Brumer-Stark). There exists u L ( r ) U S , T u ∈ L ( r ) U S ′ , T u inL^((r))U_(S^('),T)u \in \mathscr{L}^{(r)} U_{S^{\prime}, T}u∈L(r)US′,T such that
φ ( λ Q ( u ) ) = Θ S , T φ λ Q ( u ) = Θ S , T varphi(lambda_(Q)(u))=Theta_(S,T)\varphi\left(\lambda_{\mathbf{Q}}(u)\right)=\Theta_{S, T}φ(λQ(u))=ΘS,T
We describe in Theorem 4.3 below a strong partial result toward this conjecture.

4. THE BRUMER-STARK CONJECTURE

Having stated the higher rank Rubin-Brumer-Stark conjecture, we now wind back the clock and focus on the case r = 1 r = 1 r=1r=1r=1. This case had been studied independently by Brumer and Stark and served as a motivation for Rubin's work. Writing the splitting prime v 1 v 1 v_(1)v_{1}v1 as p p p\mathfrak{p}p, the conjecture may be stated as follows.
Conjecture 4.1 (Brumer-Stark). Fix a prime ideal p O F , p S T p ⊂ O F , p ∉ S ∪ T psubO_(F),p!in S uu T\mathfrak{p} \subset O_{F}, \mathfrak{p} \notin S \cup Tp⊂OF,p∉S∪T, such that p p p\mathfrak{p}p splits completely in H H HHH. Fix a prime P O H P ⊂ O H PsubO_(H)\mathfrak{P} \subset O_{H}P⊂OH above p p p\mathfrak{p}p. There exists a unique element u p H u p ∈ H ∗ u_(p)inH^(**)u_{\mathfrak{p}} \in H^{*}up∈H∗ such that | u p | w = 1 u p w = 1 |u_(p)|_(w)=1\left|u_{\mathfrak{p}}\right|_{w}=1|up|w=1 for every place w w www of H H HHH not lying above p p p\mathfrak{p}p,
ord G ( u p ) := σ G ord β ( σ ( u p ) ) σ 1 = Θ S , T ord G ⁡ u p := ∑ σ ∈ G   ord β ⁡ σ u p σ − 1 = Θ S , T ord_(G)(u_(p)):=sum_(sigma in G)ord_(beta)(sigma(u_(p)))sigma^(-1)=Theta_(S,T)\operatorname{ord}_{G}\left(u_{\mathfrak{p}}\right):=\sum_{\sigma \in G} \operatorname{ord}_{\mathfrak{\beta}}\left(\sigma\left(u_{\mathfrak{p}}\right)\right) \sigma^{-1}=\Theta_{S, T}ordG⁡(up):=∑σ∈Gordβ⁡(σ(up))σ−1=ΘS,T
and u 1 ( mod q ) u ≡ 1 ( mod q ) u-=1(modq)u \equiv 1(\bmod \mathfrak{q})u≡1(modq) for all q T H q ∈ T H qinT_(H)\mathfrak{q} \in T_{H}q∈TH.
Note that the condition | u | w = 1 | u | w = 1 |u|_(w)=1|u|_{w}=1|u|w=1 includes all complex places w w www, so c ( u p ) = u p 1 c u p = u p − 1 c(u_(p))=u_(p)^(-1)c\left(u_{\mathfrak{p}}\right)=u_{\mathfrak{p}}^{-1}c(up)=up−1 for the unique complex conjugation c G c ∈ G c in Gc \in Gc∈G.
As we have alread noted, Stark arrived upon this statement in the 1970s through his attempts to generalize and factorize the classical Dirichlet class number formula (though in a slightly different formulation; the statement above is due to Tate [46]). Prior to this, in the 1960s, Brumer was interested in generalizing Stickelberger's classical factorization formula for Gauss sums in cyclotomic fields. Stickelberger's result can be formulated as stating that when H = Q ( μ N ) H = Q μ N H=Q(mu_(N))H=\mathbf{Q}\left(\mu_{N}\right)H=Q(μN) is a cyclotomic field, the Stickelberger element annihilates the class group of H H HHH. Let us consider Brumer's perspective of annihilation of class groups in the case of general H / F H / F H//FH / FH/F.

4.1. Annihilation of class groups

Let C l T ( H ) C l T ( H ) Cl^(T)(H)\mathrm{Cl}^{T}(H)ClT(H) denote the ray class group of H H HHH with conductor equal to the product of primes in T H T H T_(H)T_{H}TH. This is defined as follows. Let I T ( H ) I T ( H ) I_(T)(H)I_{T}(H)IT(H) denote the group of fractional ideals of H H HHH relatively prime to the primes in T H T H T_(H)T_{H}TH. Let P T ( H ) P T ( H ) P_(T)(H)P_{T}(H)PT(H) denote the subgroup of I T ( H ) I T ( H ) I_(T)(H)I_{T}(H)IT(H) generated by principal ideals ( α ) ( α ) (alpha)(\alpha)(α) where α O H α ∈ O H alpha inO_(H)\alpha \in O_{H}α∈OH satisfies α 1 ( mod q ) α ≡ 1 ( mod q ) alpha-=1(mod q)\alpha \equiv 1(\bmod q)α≡1(modq) for all q T H q ∈ T H q inT_(H)q \in T_{H}q∈TH. Then
C l T ( H ) = I T ( H ) / P T ( H ) C l T ( H ) = I T ( H ) / P T ( H ) Cl^(T)(H)=I_(T)(H)//P_(T)(H)\mathrm{Cl}^{T}(H)=I_{T}(H) / P_{T}(H)ClT(H)=IT(H)/PT(H)
This T T TTT-smoothed class group is naturally a G G GGG-module.
With the notation as in Conjecture 4.1, we have
(4.2) B Θ S , T = ( u p ) (4.2) B Θ S , T = u p {:(4.2)B^(Theta_(S,T))=(u_(p)):}\begin{equation*} \mathfrak{B}^{\Theta_{S, T}}=\left(u_{\mathfrak{p}}\right) \tag{4.2} \end{equation*}(4.2)BΘS,T=(up)
Such an equation holds for all p S T p ∉ S ∪ T p!in S uu Tp \notin S \cup Tp∉S∪T that split completely in H H HHH. The set of primes of H H HHH above all such p p p\mathfrak{p}p generate C l T ( H ) C l T ( H ) Cl^(T)(H)\mathrm{Cl}^{T}(H)ClT(H). Hence we deduce
(4.3) Θ S , T Ann Z [ G ] ( C l T ( H ) ) (4.3) Θ S , T ∈ Ann Z [ G ] ⁡ C l T ( H ) {:(4.3)Theta_(S,T)inAnn_(Z[G])(Cl^(T)(H)):}\begin{equation*} \Theta_{S, T} \in \operatorname{Ann}_{\mathbf{Z}[G]}\left(\mathrm{Cl}^{T}(H)\right) \tag{4.3} \end{equation*}(4.3)ΘS,T∈AnnZ[G]⁡(ClT(H))
In fact, (4.3) is almost equivalent to Conjecture 4.1; given (4.2), the element u p u p u_(p)u_{\mathfrak{p}}up satisfies all of the conditions necessary for Conjecture 4.1 except possibly c ( u p ) = u p 1 c u p = u p − 1 c(u_(p))=u_(p)^(-1)c\left(u_{\mathfrak{p}}\right)=u_{\mathfrak{p}}^{-1}c(up)=up−1. But of course v p = u p / c ( u p ) v p = u p / c u p v_(p)=u_(p)//c(u_(p))v_{\mathfrak{p}}=u_{\mathfrak{p}} / c\left(u_{\mathfrak{p}}\right)vp=up/c(up) satisfies this condition and moreover satisfies P 2 Θ S , T = ( v p ) P 2 Θ S , T = v p P^(2Theta_(S,T))=(v_(p))\mathfrak{P}^{2 \Theta_{S, T}}=\left(v_{\mathfrak{p}}\right)P2ΘS,T=(vp). Therefore the
only possible discrepancy between the statements is a factor of 2 , which disappears when we localize away from 2 as in the rest of this paper. Let us therefore define
R = Z [ 1 / 2 ] [ G ] = Z [ 1 / 2 ] [ G ] / ( c + 1 ) R = Z [ 1 / 2 ] [ G ] − = Z [ 1 / 2 ] [ G ] / ( c + 1 ) R=Z[1//2][G]^(-)=Z[1//2][G]//(c+1)R=\mathbf{Z}[1 / 2][G]^{-}=\mathbf{Z}[1 / 2][G] /(c+1)R=Z[1/2][G]−=Z[1/2][G]/(c+1)
and for any Z [ G ] Z [ G ] Z[G]\mathbf{Z}[G]Z[G]-module M M MMM we write M = M Z [ G ] R M − = M ⊗ Z [ G ] R M^(-)=Mox_(Z[G])RM^{-}=M \otimes_{\mathbf{Z}[G]} RM−=M⊗Z[G]R. There exists an element u p O H [ 1 / p ] Z [ 1 / 2 ] u p ∈ O H [ 1 / p ] ∗ ⊗ Z [ 1 / 2 ] u_(p)inO_(H)[1//p]^(**)oxZ[1//2]u_{\mathfrak{p}} \in O_{H}[1 / \mathfrak{p}]^{*} \otimes \mathbf{Z}[1 / 2]up∈OH[1/p]∗⊗Z[1/2] satisfying Conjecture 4.1 if and only if
(4.4) Θ S , T Ann R ( C l T ( H ) ) (4.4) Θ S , T ∈ Ann R ⁡ C l T ( H ) − {:(4.4)Theta_(S,T)inAnn_(R)(Cl^(T)(H)^(-)):}\begin{equation*} \Theta_{S, T} \in \operatorname{Ann}_{R}\left(\mathrm{Cl}^{T}(H)^{-}\right) \tag{4.4} \end{equation*}(4.4)ΘS,T∈AnnR⁡(ClT(H)−)
This is the Brumer-Stark conjecture "away from 2".
Many authors have studied (4.4) as well as refinements. The works of Burns, Greither, Kurihara, Popescu, and Sano are particularly noteworthy [4-7, 25-27,35]. Many of these refinements involve Fitting ideals, whose definition we now recall.
Let R R RRR be a commutative ring and M M MMM an R R RRR-module with finite presentation:
R m A R n X 0 R m → A R n → X → 0 R^(m)rarr"A"R^(n)rarr X rarr0R^{m} \xrightarrow{A} R^{n} \rightarrow X \rightarrow 0Rm→ARn→X→0
Here A A AAA is an n × m n × m n xx mn \times mn×m matrix over R R RRR. The i i iii th Fitting ideal Fitt i , R ( M ) i , R ( M ) i,R(M)i, R(M)i,R(M) is the ideal generated by the n i × n i n − i × n − i n-i xx n-in-i \times n-in−i×n−i minors of A A AAA. It is a standard fact [37, CHAPTER 3, THEOREM 1] that Fitt i , R ( M ) i , R ( M ) _(i,R)(M){ }_{i, R}(M)i,R(M) does not depend on the chosen presentation of M M MMM. We write Fitt R ( M ) Fitt R ⁡ ( M ) Fitt_(R)(M)\operatorname{Fitt}_{R}(M)FittR⁡(M) for Fitt 0 , R ( M ) Fitt 0 , R ⁡ ( M ) Fitt_(0,R)(M)\operatorname{Fitt}_{0, R}(M)Fitt0,R⁡(M), and when these is no ambiguity about the choice of i = 0 i = 0 i=0i=0i=0, we call this the Fitting ideal of M M MMM.
The Fitting ideal of a finitely presented module is contained in its annihilator:
(4.5) Fitt R ( M ) Ann R ( M ) (4.5) Fitt R ⁡ ( M ) ⊂ Ann R ⁡ ( M ) {:(4.5)Fitt_(R)(M)subAnn_(R)(M):}\begin{equation*} \operatorname{Fitt}_{R}(M) \subset \operatorname{Ann}_{R}(M) \tag{4.5} \end{equation*}(4.5)FittR⁡(M)⊂AnnR⁡(M)
In view of (4.4) and (4.5), it is therefore natural to ask whether Θ S , T Θ S , T Theta_(S,T)\Theta_{S, T}ΘS,T lies in the Fitting ideal of C l T ( H ) C l T ( H ) − Cl^(T)(H)^(-)\mathrm{Cl}^{T}(H)^{-}ClT(H)−over R R RRR. It was noticed by Popescu in the function field case [38] and by Kurihara in the number field case that while this holds sometimes, it does not always hold. Greither and Kurihara [ 25 , 26 ] [ 25 , 26 ] [25,26][25,26][25,26] observed that the statement may be corrected by replacing C l T ( H ) C l T ( H ) − Cl^(T)(H)^(-)\mathrm{Cl}^{T}(H)^{-}ClT(H)−by its Pontryagin dual
C l T ( H ) , = Hom Z ( C l T ( H ) , Q / Z ) C l T ( H ) − , ∨ = Hom Z ⁡ C l T ( H ) − , Q / Z Cl^(T)(H)^(-,vv)=Hom_(Z)(Cl^(T)(H)^(-),Q//Z)\mathrm{Cl}^{T}(H)^{-, \vee}=\operatorname{Hom}_{\mathbf{Z}}\left(\mathrm{Cl}^{T}(H)^{-}, \mathbf{Q} / \mathbf{Z}\right)ClT(H)−,∨=HomZ⁡(ClT(H)−,Q/Z)
We endow C l T ( H ) , C l T ( H ) − , ∨ Cl^(T)(H)^(-,vv)\mathrm{Cl}^{T}(H)^{-, \vee}ClT(H)−,∨ with the contragradient G G GGG-action g φ ( x ) = φ ( g 1 x ) g â‹… φ ( x ) = φ g − 1 x g*varphi(x)=varphi(g^(-1)x)g \cdot \varphi(x)=\varphi\left(g^{-1} x\right)g⋅φ(x)=φ(g−1x). Denote by # the involution on Z [ G ] Z [ G ] Z[G]\mathbf{Z}[G]Z[G] induced by g g 1 g ↦ g − 1 g|->g^(-1)g \mapsto g^{-1}g↦g−1 for g G g ∈ G g in Gg \in Gg∈G.
Conjecture 4.2 (Kurihara, "Strong Brumer-Stark"). We have
Θ S , T # Fitt R ( C l T ( H ) , v ) Θ S , T # ∈ Fitt R ⁡ C l T ( H ) − , v Theta_(S,T)^(#)inFitt_(R)(Cl^(T)(H)^(-,v))\Theta_{S, T}^{\#} \in \operatorname{Fitt}_{R}\left(\mathrm{Cl}^{T}(H)^{-, v}\right)ΘS,T#∈FittR⁡(ClT(H)−,v)
Conjecture 4.2 leads to the following natural questions:
(1) What is the Fitting ideal of C l T ( H ) , v C l T ( H ) − , v Cl^(T)(H)^(-,v)\mathrm{Cl}^{T}(H)^{-, v}ClT(H)−,v ?
(2) What is the Fitting ideal of C l T ( H ) C l T ( H ) − Cl^(T)(H)^(-)\mathrm{Cl}^{T}(H)^{-}ClT(H)−?
(3) Is there a natural arithmetically defined R R RRR-module whose Fitting ideal is generated by Θ S , T Θ S , T Theta_(S,T)\Theta_{S, T}ΘS,T or Θ S , T # Θ S , T # Theta_(S,T)^(#)\Theta_{S, T}^{\#}ΘS,T# ?
The precise conjectural description of the Fitting ideal of C l T ( H / F ) , C l T ( H / F ) − , ∨ Cl^(T)(H//F)^(-,vv)\mathrm{Cl}^{T}(H / F)^{-, \vee}ClT(H/F)−,∨ was given by Kurihara [35]; we state this in Section 5.1 below. An important fact about this statement is that when S ram S ram  S_("ram ")S_{\text {ram }}Sram  is nonempty, the Fitting ideal of C l T ( H / F ) , C l T ( H / F ) − , ∨ Cl^(T)(H//F)^(-,vv)\mathrm{Cl}^{T}(H / F)^{-, \vee}ClT(H/F)−,∨ is in general not principal (and in particular is not generated by Θ S , T # Θ S , T # Theta_(S,T)^(#)\Theta_{S, T}^{\#}ΘS,T# ).
A conjectural answer to the second question above has recently been provided in a striking paper by Atsuta and Kataoka [1]. They show that their conjecture is implied by the Equivariant Tamagawa Number Conjecture.
The third question is answered by a conjecture of Burns, Kurihara, and Sano, and is the topic of Section § 5.3 § 5.3 §5.3\S 5.3§5.3. We note that Fitting ideals of finitely presented R R RRR-modules are rarely principal. It is therefore remarkable that Burns-Kurihara-Sano defined a natural arithmetic object whose Fitting ideal is principal.

4.2. Our results

We now describe some of our results toward these conjectures [17, THEOREM 1.4].
Theorem 4.3. Kurihara's exact formula for F i t t R ( C l T ( H ) , ) F i t t R C l T ( H ) − , ∨ Fitt_(R)(Cl^(T)(H)^(-,vv))\mathrm{Fitt}_{R}\left(\mathrm{Cl}^{T}(H)^{-, \vee}\right)FittR(ClT(H)−,∨) holds (see Theorem 5.1). In particular, we have the Brumer-Stark and Strong Brumer-Stark conjectures away from 2:
Θ S , T # Fitt R ( C l T ( H ) , v ) Ann R ( C l T ( H ) ) # Θ S , T # ∈ Fitt R ⁡ C l T ( H ) − , v ⊂ Ann R ⁡ C l T ( H ) − # Theta_(S,T)^(#)inFitt_(R)(Cl^(T)(H)^(-,v))subAnn_(R)(Cl^(T)(H)^(-))^(#)\Theta_{S, T}^{\#} \in \operatorname{Fitt}_{R}\left(\mathrm{Cl}^{T}(H)^{-, v}\right) \subset \operatorname{Ann}_{R}\left(\mathrm{Cl}^{T}(H)^{-}\right)^{\#}ΘS,T#∈FittR⁡(ClT(H)−,v)⊂AnnR⁡(ClT(H)−)#
Finally, Rubin's higher rank Brumer-Stark conjecture holds away from 2: with notation as in Conjecture 3.3, there exists u L ( r ) U S , T Z [ 1 / 2 ] u ∈ L ( r ) U S ′ , T ⊗ Z [ 1 / 2 ] u inL^((r))U_(S^('),T)oxZ[1//2]u \in \mathscr{L}^{(r)} U_{S^{\prime}, T} \otimes \mathbf{Z}[1 / 2]u∈L(r)US′,T⊗Z[1/2] such that φ ( λ Q ( u ) ) = Θ S , T φ λ Q ( u ) = Θ S , T varphi(lambda_(Q)(u))=Theta_(S,T)\varphi\left(\lambda_{\mathbf{Q}}(u)\right)=\Theta_{S, T}φ(λQ(u))=ΘS,T.
Partial results in this direction had been obtained earlier by Burns [5] (including a μ = 0 μ = 0 mu=0\mu=0μ=0 hypothesis and the assumption of the Gross-Kuz'min conjecture) and by Greither and Popescu [27] (including a μ = 0 μ = 0 mu=0\mu=0μ=0 hypothesis and imprimitivity conditions on S S SSS ).
Our results in [17] do not seem to directly imply the conjecture of Atsuta and Kataoka on Fitt R ( C l T ( H ) ) R C l T ( H ) − _(R)(Cl^(T)(H)^(-)){ }_{R}\left(\mathrm{Cl}^{T}(H)^{-}\right)R(ClT(H)−)or the conjecture of Burns. However, we prove an analogous result toward the latter, with ( S , T ) ( S , T ) (S,T)(S, T)(S,T) replaced by an alternate pair ( Σ , Σ ) Σ , Σ ′ (Sigma,Sigma^('))\left(\Sigma, \Sigma^{\prime}\right)(Σ,Σ′), in Theorem 5.6. This result turns out to be strong enough to deduce Theorem 4.3.
In § 6 § 6 §6\S 6§6 we give a detailed summary of the proof of Theorem 5.6. Key ingredients are the Z [ G ] Z [ G ] Z[G]\mathbf{Z}[G]Z[G]-modules Σ Σ ( H ) ∇ Σ Σ ′ ( H ) grad_(Sigma)^(Sigma^('))(H)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)∇ΣΣ′(H) defined by Ritter and Weiss, and Ribet's method of using modular forms to construct Galois cohomology classes associated to L L LLL-functions.

4.3. Explicit formula for Brumer-Stark units

We conclude this section by describing a further direction in the study of BrumerStark units, that of explicit formulae and applications to explicit class field theory. This theme was initiated by Gross in [28] and [29] and developed by the first author and collaborators over a series of papers [ 9 , 10 , 13 , 15 , 20 ] [ 9 , 10 , 13 , 15 , 20 ] [9,10,13,15,20][9,10,13,15,20][9,10,13,15,20].
Let p p p\mathfrak{p}p be as above and write S = S { p } S ′ = S ∪ { p } S^(')=S uu{p}S^{\prime}=S \cup\{\mathfrak{p}\}S′=S∪{p}. Let L L LLL denote a finite abelian CM extension of F F FFF containing H H HHH that is ramified over F F FFF only at the places in S S ′ S^(')S^{\prime}S′. Write g = Gal ( L / F ) g = Gal ⁡ ( L / F ) g=Gal(L//F)\mathrm{g}=\operatorname{Gal}(L / F)g=Gal⁡(L/F) and Γ = Gal ( L / H ) Γ = Gal ⁡ ( L / H ) Gamma=Gal(L//H)\Gamma=\operatorname{Gal}(L / H)Γ=Gal⁡(L/H), so g / Γ G g / Γ ≅ G g//Gamma~=G\mathrm{g} / \Gamma \cong Gg/Γ≅G. Let I I III denote the relative augmentation ideal associated to g g ggg and G G GGG, i.e., the kernel of the canonical projection Aug: Z [ g ] Z [ G ] Z [ g ] → Z [ G ] Z[g]rarrZ[G]\mathbf{Z}[\mathfrak{g}] \rightarrow \mathbf{Z}[G]Z[g]→Z[G]. Then Θ S , T ( L / F ) Θ S ′ , T ( L / F ) Theta_(S^('),T)(L//F)\Theta_{S^{\prime}, T}(L / F)ΘS′,T(L/F)
lies in I I III, since its image under Aug is
(4.6) Θ S , T ( H / F ) = Θ S , T ( H / F ) ( 1 σ p ) = 0 (4.6) Θ S ′ , T ( H / F ) = Θ S , T ( H / F ) 1 − σ p = 0 {:(4.6)Theta_(S^('),T)(H//F)=Theta_(S,T)(H//F)(1-sigma_(p))=0:}\begin{equation*} \Theta_{S^{\prime}, T}(H / F)=\Theta_{S, T}(H / F)\left(1-\sigma_{\mathfrak{p}}\right)=0 \tag{4.6} \end{equation*}(4.6)ΘS′,T(H/F)=ΘS,T(H/F)(1−σp)=0
Here σ p σ p sigma_(p)\sigma_{\mathfrak{p}}σp denotes the Frobenius associated to p p p\mathfrak{p}p in G G GGG, and this is trivial since p p p\mathfrak{p}p splits completely in H H HHH. Intuitively, if we view Θ S , T ( L / F ) Θ S ′ , T ( L / F ) Theta_(S^('),T)(L//F)\Theta_{S^{\prime}, T}(L / F)ΘS′,T(L/F) as a function on the ideals of Z [ g ] Z [ g ] Z[g]\mathbf{Z}[\mathrm{g}]Z[g], equation (4.6) states that this function "has a zero" at the ideal I I III; the value of the "derivative" of this function at I I III is simply the image of Θ S , T ( L / F ) Θ S ′ , T ( L / F ) Theta_(S^('),T)(L//F)\Theta_{S^{\prime}, T}(L / F)ΘS′,T(L/F) in I / I 2 I / I 2 I//I^(2)I / I^{2}I/I2. Gross provided a conjectural algebraic interpretation of this derivative as follows. Denote by
rec P : H B Γ rec P : H B ∗ → Γ rec_(P):H_(B)^(**)rarr Gamma\operatorname{rec}_{\mathfrak{P}}: H_{\mathfrak{B}}^{*} \rightarrow \GammarecP:HB∗→Γ
the composition of the inclusion H ß A H H ß ∗ ↪ A H ∗ H_(ß)^(**)↪A_(H)^(**)H_{\mathfrak{ß}}^{*} \hookrightarrow \mathbf{A}_{H}^{*}Hß∗↪AH∗ with the global Artin reciprocity map
A H Γ A H ∗ → Γ A_(H)^(**)rarr Gamma\mathbf{A}_{H}^{*} \rightarrow \GammaAH∗→Γ
Throughout this article we adopt Serre's convention [42] for the reciprocity map. Therefore rec ( ϖ 1 ) rec ⁡ Ï– − 1 rec(Ï–^(-1))\operatorname{rec}\left(\varpi^{-1}\right)rec⁡(ϖ−1) is a lifting to G p a b G p a b G_(p)^(ab)G_{\mathfrak{p}}^{\mathrm{ab}}Gpab of the Frobenius element on the maximal unramified extension of F p F p F_(p)F_{\mathfrak{p}}Fp if ϖ F p Ï– ∈ F p ∗ Ï–inF_(p)^(**)\varpi \in F_{\mathfrak{p}}^{*}ϖ∈Fp∗ is a uniformizer.
Conjecture 4.4 (Gross, [29, CONJEcTURE 7.6]). Define
(4.7) rec G ( u p ) = σ G ( rec p σ ( u p ) 1 ) σ ~ 1 I / I 2 (4.7) rec G ⁡ u p = ∑ σ ∈ G   rec p ⁡ σ u p − 1 σ ~ − 1 ∈ I / I 2 {:(4.7)rec_(G)(u_(p))=sum_(sigma in G)(rec_(p)sigma(u_(p))-1) tilde(sigma)^(-1)in I//I^(2):}\begin{equation*} \operatorname{rec}_{G}\left(u_{\mathfrak{p}}\right)=\sum_{\sigma \in G}\left(\operatorname{rec}_{\mathfrak{p}} \sigma\left(u_{\mathfrak{p}}\right)-1\right) \tilde{\sigma}^{-1} \in I / I^{2} \tag{4.7} \end{equation*}(4.7)recG⁡(up)=∑σ∈G(recp⁡σ(up)−1)σ~−1∈I/I2
where σ ~ g σ ~ ∈ g tilde(sigma)ing\tilde{\sigma} \in \mathrm{g}σ~∈g is any lift of σ G σ ∈ G sigma in G\sigma \in Gσ∈G. Then
rec G ( u p ) Θ S , T L / F in I / I 2 rec G ⁡ u p ≡ Θ S ′ , T L / F  in  I / I 2 rec_(G)(u_(p))-=Theta_(S^('),T)^(L//F)quad" in "I//I^(2)\operatorname{rec}_{G}\left(u_{\mathfrak{p}}\right) \equiv \Theta_{S^{\prime}, T}^{L / F} \quad \text { in } I / I^{2}recG⁡(up)≡ΘS′,TL/F in I/I2
The main result of [18] is the following.
Theorem 4.5. Let p p ppp be an odd prime and suppose that p p p\mathrm{p}p lies above p. Gross's Conjecture 4.4 holds in ( I / I 2 ) Z p I / I 2 ⊗ Z p (I//I^(2))oxZ_(p)\left(I / I^{2}\right) \otimes \mathbf{Z}_{p}(I/I2)⊗Zp.
Our interest in this result is that by enlarging S S SSS and taking larger and larger field extensions L / F L / F L//FL / FL/F, one can use (4.7) to specify all of the p p p\mathfrak{p}p-adic digits of u p u p u_(p)u_{\mathfrak{p}}up. One therefore obtains an exact p p p\mathfrak{p}p-adic analytic formula for u p u p u_(p)u_{\mathfrak{p}}up. This formula can be described either using the Eisenstein cocycle or more explicitly via Shintani's method; the latter approach is followed in Section 7.2. In Section 7.3, we describe the argument using "horizontal Iwasawa theory" to show that Theorem 4.5 implies the conjectural exact formula. In Section 7.4 we summarize the key ingredients involved in the proof of Theorem 4.5, including an integral version of the Greenberg-Stevens L L L\mathscr{L}L-invariant and an associated modified Ritter-Weiss module L ∇ L grad_(L)\nabla_{\mathscr{L}}∇L. In the setting of F F FFF real quadratic, Darmon, Pozzi, and Vonk have given an alternate, elegant proof of the explicit formula for the units u p u p u_(p)u_{\mathfrak{p}}up (Section 7.5). Their approach involves p p ppp-adic deformations of modular forms, rather than the tame deformations that we consider.
One significance of the exact formula is that we show that the collection of BrumerStark units, together with some easily described elements, generate the maximal abelian extension of the totally real field F F FFF.
Theorem 4.6. Let BS denote the set of Brumer-Stark units u p u p u_(p)u_{\mathfrak{p}}up as we range over all possible CMabelian extensions H / F H / F H//FH / FH/F and for each extension a choice of prime p p ppp that splits completely in H H HHH. Let { α 1 , , α n 1 } α 1 , … , α n − 1 {alpha_(1),dots,alpha_(n-1)}\left\{\alpha_{1}, \ldots, \alpha_{n-1}\right\}{α1,…,αn−1} denote any elements of F F ∗ F^(**)F^{*}F∗ whose signs in { ± 1 } n / ( 1 , , 1 ) { ± 1 } n / ( − 1 , … , − 1 ) {+-1}^(n)//(-1,dots,-1)\{ \pm 1\}^{n} /(-1, \ldots,-1){±1}n/(−1,…,−1) under the real embeddings of F F FFF form a basis for this Z / 2 Z Z / 2 Z Z//2Z\mathbf{Z} / 2 \mathbf{Z}Z/2Z-vector space. The maximal abelian extension of F F FFF is generated by B S B S BS\mathrm{BS}BS together with α 1 , , α n 1 α 1 , … , α n − 1 sqrt(alpha_(1)),dots,sqrt(alpha_(n-1))\sqrt{\alpha_{1}}, \ldots, \sqrt{\alpha_{n-1}}α1,…,αn−1 :
F a b = F ( B S , α 1 , , α n 1 ) F a b = F B S , α 1 , … , α n − 1 F^(ab)=F(BS,sqrt(alpha_(1)),dots,sqrt(alpha_(n-1)))F^{\mathrm{ab}}=F\left(\mathrm{BS}, \sqrt{\alpha_{1}}, \ldots, \sqrt{\alpha_{n-1}}\right)Fab=F(BS,α1,…,αn−1)
It is important to stress that the exact formula for u p u p u_(p)u_{\mathfrak{p}}up described in Sectin 7.2 can be computed without knowledge of the field H H HHH, using only the data of F F FFF, p, and the conductor of H / F H / F H//FH / FH/F. Furthermore, we can leave out any finite set of primes p p p\mathfrak{p}p without altering the conclusion of the theorem. In this way we obtain an explicit class field theory for F F FFF, i.e., an analytic construction of its maximal abelian extension F ab F ab  F^("ab ")F^{\text {ab }}Fab  using data intrinsic only to F F FFF itself. Explicit computations of class fields of real quadratic fields generated using our formula are provided in [17, §2.3] and [23].

5. REFINEMENTS OF STARK'S CONJECTURE

In this section we recall various refinements of the strong Brumer-Stark conjecture.

5.1. The conjecture of Kurihara

In this section we describe the Fitting ideal of the minus part of the dual class group. For each v v vvv in S ram S ram  S_("ram ")S_{\text {ram }}Sram , let I v G v G I v ⊂ G v ⊂ G I_(v)subG_(v)sub GI_{v} \subset G_{v} \subset GIv⊂Gv⊂G denote the inertia and decomposition groups, respectively, associated to v v vvv. Write
e v = 1 # I v N I v = 1 # I v σ I v σ Q [ G ] e v = 1 # I v N I v = 1 # I v ∑ σ ∈ I v   σ ∈ Q [ G ] e_(v)=(1)/(#I_(v))NI_(v)=(1)/(#I_(v))sum_(sigma inI_(v))sigma inQ[G]e_{v}=\frac{1}{\# I_{v}} \mathrm{~N} I_{v}=\frac{1}{\# I_{v}} \sum_{\sigma \in I_{v}} \sigma \in \mathbf{Q}[G]ev=1#Iv NIv=1#Iv∑σ∈Ivσ∈Q[G]
for the idempotent that represents projection onto the characters of G G GGG unramified at v v vvv. Let σ v G v σ v ∈ G v sigma_(v)inG_(v)\sigma_{v} \in G_{v}σv∈Gv denote any representative of the Frobenius coset of v v vvv. The element 1 σ v e v Q [ G ] 1 − σ v e v ∈ Q [ G ] 1-sigma_(v)e_(v)inQ[G]1-\sigma_{v} e_{v} \in \mathbf{Q}[G]1−σvev∈Q[G] is independent of choice of representative. Following [25], we define the Sinnott-Kurihara ideal, a priori a fractional ideal of Z [ G ] Z [ G ] Z[G]\mathbf{Z}[G]Z[G], by
S K u T ( H / F ) = ( Θ S , T # ) v S r a m ( N I v , 1 σ v e v ) S K u T ( H / F ) = Θ S ∞ , T # ∏ v ∈ S r a m   N I v , 1 − σ v e v SKu^(T)(H//F)=(Theta_(S_(oo),T)^(#))prod_(v inS_(ram))(NI_(v),1-sigma_(v)e_(v))\mathrm{SKu}^{T}(H / F)=\left(\Theta_{S_{\infty}, T}^{\#}\right) \prod_{v \in S_{\mathrm{ram}}}\left(\mathrm{N} I_{v}, 1-\sigma_{v} e_{v}\right)SKuT(H/F)=(ΘS∞,T#)∏v∈Sram(NIv,1−σvev)
Kurihara proved using the theorem of Deligne-Ribet and Cassou-Noguès that S K u T ( H / F ) S K u T ( H / F ) SKu^(T)(H//F)\mathrm{SKu}^{T}(H / F)SKuT(H/F) is a subset of Z [ G ] Z [ G ] Z[G]\mathbf{Z}[G]Z[G] (see [17, LEMMA 3.4]). The following conjecture of Kurihara is proven in [ 17 [ 17 [17[17[17, THEOREM 1.4].
Theorem 5.1. We have
Fitt R ( C l T ( H ) , v ) = S K u T ( H / F ) Fitt R ⁡ C l T ( H ) − , v = S K u T ( H / F ) − Fitt_(R)(Cl^(T)(H)^(-,v))=SKu^(T)(H//F)^(-)\operatorname{Fitt}_{R}\left(\mathrm{Cl}^{T}(H)^{-, v}\right)=\mathrm{SKu}^{T}(H / F)^{-}FittR⁡(ClT(H)−,v)=SKuT(H/F)−
The definition of the Sinnott-Kurihara ideal is inspired by Sinnott's definition of generalized Stickelberger elements for abelian extensions of Q Q Q\mathbf{Q}Q [44]. For a generalization of Sinnott's ideal to arbitrary totally real fields see [25, $2]. In general, Sinnott's ideal contains the Sinnott-Kurihara ideal but it may be strictly larger.
The plus part of the Sinnott-Kurihara ideal is not very interesting as the plus part of Θ S , T # Θ S ∞ , T # Theta_(S_(oo),T)^(#)\Theta_{S_{\infty}, T}^{\#}ΘS∞,T# is 0 . The plus part of the class group is much smaller than the minus part and seems harder to describe; for example, Greenberg's conjecture on the vanishing of lambda invariants implies that the order of the plus part is bounded up the cyclotomic tower. For abelian extensions of Q Q Q\mathbf{Q}Q, the plus part is described by Sinnott using cyclotomic units.

5.2. The conjecture of Atsuta-Kataoka

It is, in fact, more natural to ask about the Fitting ideal of C l T ( H ) C l T ( H ) Cl^(T)(H)\mathrm{Cl}^{T}(H)ClT(H), as opposed to the Pontryagin dual. A conjectural answer to this question has been provided in a recent paper of Atsuta-Kataoka [1] using the theory of shifted Fitting ideals developed by Kataoka [32]. We recall this notion now.
Let M M MMM be an R R RRR-module of finite length. Take a resolution
0 N P 1 P d M 0 0 → N → P 1 → ⋯ → P d → M → 0 0rarr N rarrP_(1)rarr cdots rarrP_(d)rarr M rarr00 \rightarrow N \rightarrow P_{1} \rightarrow \cdots \rightarrow P_{d} \rightarrow M \rightarrow 00→N→P1→⋯→Pd→M→0
with each P i P i P_(i)P_{i}Pi of projective dimension 1 ≤ 1 <= 1\leq 1≤1. Following [32], define the shifted Fitting ideal
Fitt R [ d ] ( M ) = ( i = 1 d Fitt R ( P i ) ( 1 ) i ) Fitt R ( N ) Fitt R [ d ] ⁡ ( M ) = ∏ i = 1 d   Fitt R ⁡ P i ( − 1 ) i Fitt R ⁡ ( N ) Fitt_(R)^([d])(M)=(prod_(i=1)^(d)Fitt_(R)(P_(i))^((-1)^(i)))Fitt_(R)(N)\operatorname{Fitt}_{R}^{[d]}(M)=\left(\prod_{i=1}^{d} \operatorname{Fitt}_{R}\left(P_{i}\right)^{(-1)^{i}}\right) \operatorname{Fitt}_{R}(N)FittR[d]⁡(M)=(∏i=1dFittR⁡(Pi)(−1)i)FittR⁡(N)
The independence of this definition from the choice of resolution is proven in [32, THEOREM 2.6 AND PROPOSITION 2.7]. Let
g v = 1 σ v + # I v Z [ G / I v ] , h v = 1 e v σ v + N I v Q [ G ] g v = 1 − σ v + # I v ∈ Z G / I v , h v = 1 − e v σ v + N I v ∈ Q [ G ] g_(v)=1-sigma_(v)+#I_(v)inZ[G//I_(v)],quadh_(v)=1-e_(v)sigma_(v)+NI_(v)inQ[G]g_{v}=1-\sigma_{v}+\# I_{v} \in \mathbf{Z}\left[G / I_{v}\right], \quad h_{v}=1-e_{v} \sigma_{v}+\mathbf{N} I_{v} \in \mathbf{Q}[G]gv=1−σv+#Iv∈Z[G/Iv],hv=1−evσv+NIv∈Q[G]
Define the Z [ G ] Z [ G ] Z[G]\mathbf{Z}[G]Z[G]-module
A v = Z [ G / I v ] / ( g v ) A v = Z G / I v / g v A_(v)=Z[G//I_(v)]//(g_(v))A_{v}=\mathbf{Z}\left[G / I_{v}\right] /\left(g_{v}\right)Av=Z[G/Iv]/(gv)
Conjecture 5.2 (Atsuta-Kataoka). We have
Fitt R ( C l T ( H ) ) = ( w S r a m , H h v Fitt R [ 1 ] ( A v ) ) Θ S , T Fitt R ⁡ C l T ( H ) − = ∏ w ∈ S r a m , H   h v − Fitt R [ 1 ] ⁡ A v − Θ S ∞ , T Fitt_(R)(Cl^(T)(H)^(-))=(prod_(w inS_(ram,H))h_(v)^(-)Fitt_(R)^([1])(A_(v)^(-)))Theta_(S_(oo),T)\operatorname{Fitt}_{R}\left(\mathrm{Cl}^{T}(H)^{-}\right)=\left(\prod_{w \in S_{\mathrm{ram}, H}} h_{v}^{-} \operatorname{Fitt}_{R}^{[1]}\left(A_{v}^{-}\right)\right) \Theta_{S_{\infty}, T}FittR⁡(ClT(H)−)=(∏w∈Sram,Hhv−FittR[1]⁡(Av−))ΘS∞,T
In [1], the authors give an explicit description of the ideal h v Fitt Z [ G ] [ 1 ] ( A v ) h v − Fitt Z [ G ] − [ 1 ] ⁡ A v − h_(v)^(-)Fitt_(Z[G]^(-))^([1])(A_(v)^(-))h_{v}^{-} \operatorname{Fitt}_{\mathrm{Z}[G]^{-}}^{[1]}\left(A_{v}^{-}\right)hv−FittZ[G]−[1]⁡(Av−)appearing in Conjecture 5.2. Write I v = J 1 × × J s I v = J 1 × ⋯ × J s I_(v)=J_(1)xx cdots xxJ_(s)I_{v}=J_{1} \times \cdots \times J_{s}Iv=J1×⋯×Js for cyclic groups J i , 1 i s J i , 1 ≤ i ≤ s J_(i),1 <= i <= sJ_{i}, 1 \leq i \leq sJi,1≤i≤s. For each i i iii, put
N i = N J i = σ J i σ Z [ G ] N i = N J i = ∑ σ ∈ J i   σ ∈ Z [ G ] N_(i)=NJ_(i)=sum_(sigma inJ_(i))sigma inZ[G]\mathrm{N}_{i}=\mathrm{N} J_{i}=\sum_{\sigma \in J_{i}} \sigma \in \mathbf{Z}[G]Ni=NJi=∑σ∈Jiσ∈Z[G]
Furthermore, put d = ker ( Z [ G ] Z [ G / G v ] ) d = ker ⁡ Z [ G ] → Z G / G v d=ker(Z[G]rarrZ[G//G_(v)])d=\operatorname{ker}\left(\mathbf{Z}[G] \rightarrow \mathbf{Z}\left[G / G_{v}\right]\right)d=ker⁡(Z[G]→Z[G/Gv]) for the relative augmentation ideal. For each 1 i s 1 ≤ i ≤ s 1 <= i <= s1 \leq i \leq s1≤i≤s, put Z i Z i Z_(i)Z_{i}Zi for the ideal of Z [ G ] Z [ G ] Z[G]\mathbf{Z}[G]Z[G] generated by N j 1 N j s i N j 1 ⋯ N j s − i N_(j_(1))cdotsN_(j_(s-i))\mathrm{N}_{j_{1}} \cdots \mathrm{N}_{j_{s-i}}Nj1⋯Njs−i, where ( j 1 , , j s i ) j 1 , … , j s − i (j_(1),dots,j_(s-i))\left(j_{1}, \ldots, j_{s-i}\right)(j1,…,js−i) runs through all tuples of integers satisfying 1 j 1 j s i s 1 ≤ j 1 ≤ ⋯ ≤ j s − i ≤ s 1 <= j_(1) <= cdots <= j_(s-i) <= s1 \leq j_{1} \leq \cdots \leq j_{s-i} \leq s1≤j1≤⋯≤js−i≤s. Define
I = i = 1 s Z i χ i 1 I = ∑ i = 1 s   Z i χ i − 1 I=sum_(i=1)^(s)Z_(i)chi^(i-1)\mathcal{I}=\sum_{i=1}^{s} Z_{i} \chi^{i-1}I=∑i=1sZiχi−1
Although Z i Z i Z_(i)Z_{i}Zi depends on the decomposition of I v I v I_(v)I_{v}Iv into cyclic groups, the ideal L L L\mathcal{L}L is independent (see [1, DEFINITION 1.2]).
Theorem 5.3 (Atsuta-Kataoka). We have
h v Fitt Z [ G ] [ 1 ] ( A v ) = ( N I v , ( 1 e v σ v ) L ) h v − Fitt Z [ G ] − [ 1 ] ⁡ A v − = N I v , 1 − e v σ v L h_(v)^(-)Fitt_(Z[G]^(-))^([1])(A_(v)^(-))=(NI_(v),(1-e_(v)sigma_(v))L)h_{v}^{-} \operatorname{Fitt}_{\mathbf{Z}[G]^{-}}^{[1]}\left(A_{v}^{-}\right)=\left(\mathrm{N} I_{v},\left(1-e_{v} \sigma_{v}\right) \mathcal{L}\right)hv−FittZ[G]−[1]⁡(Av−)=(NIv,(1−evσv)L)
as fractional ideals of Z [ G ] Z [ G ] − Z[G]^(-)\mathbf{Z}[G]^{-}Z[G]−.
Atsuta-Kataoka prove:
Theorem 5.4. The Equivariant Tamagawa Number Conjecture for H / F H / F H//FH / FH/F implies Conjecture 5.2.

5.3. The conjecture of Burns-Kurihara-Sano

The refinements mentioned above do not involve principal ideals. The method of Ribet, which attempts to show the inclusion of an arithmetically defined ideal into an analytically defined ideal, works well for principal ideals. From this point of view, it is natural to ask if there is an arithmetically defined object whose Fitting ideal is generated by the Stickelberger element Θ S , T Θ S , T Theta_(S,T)\Theta_{S, T}ΘS,T. Burns, Kurihara and Sano provided a conjectural answer to this question [6]. A modification of this statement (Theorem 5.6 below) is the main technical result in [17] from which all the results of Theorem 4.3 are deduced.
We now recall the statement of the conjecture of Burns-Kurihara-Sano. Let H T H T ∗ H_(T)^(**)H_{T}^{*}HT∗ be the group of x H x ∈ H ∗ x inH^(**)x \in H^{*}x∈H∗ such that ord w ( x 1 ) > 0 ord w ⁡ ( x − 1 ) > 0 ord_(w)(x-1) > 0\operatorname{ord}_{w}(x-1)>0ordw⁡(x−1)>0 for each prime w T H w ∈ T H w inT_(H)w \in T_{H}w∈TH. Define
Sel S T ( H ) = Hom Z ( H T , Z ) / w S H T H Z Sel S T ⁡ ( H ) = Hom Z ⁡ H T ∗ , Z / ∏ w ∉ S H ∪ T H   Z Sel_(S)^(T)(H)=Hom_(Z)(H_(T)^(**),Z)//prod_(w!inS_(H)uuT_(H))Z\operatorname{Sel}_{S}^{T}(H)=\operatorname{Hom}_{\mathbf{Z}}\left(H_{T}^{*}, \mathbf{Z}\right) / \prod_{w \notin S_{H} \cup T_{H}} \mathbf{Z}SelST⁡(H)=HomZ⁡(HT∗,Z)/∏w∉SH∪THZ
where the implicit map sends a tuple ( x w ) x w (x_(w))\left(x_{w}\right)(xw) to the function w x w ord w ∑ w   x w ord w sum_(w)x_(w)ord_(w)\sum_{w} x_{w} \operatorname{ord}_{w}∑wxwordw. The G G GGG-action on Sel S T ( H ) Sel S T ⁡ ( H ) Sel_(S)^(T)(H)\operatorname{Sel}_{S}^{T}(H)SelST⁡(H) is the contragradient G G GGG-action ( g φ ) ( x ) = φ ( g 1 x ) ( g φ ) ( x ) = φ g − 1 x (g varphi)(x)=varphi(g^(-1)x)(g \varphi)(x)=\varphi\left(g^{-1} x\right)(gφ)(x)=φ(g−1x)
Conjecture 5.5 (Burns-Kurihara-Sano). We have
Fitt R ( Sel S T ( H ) ) = ( Θ S , T # ) Fitt R ⁡ Sel S T ⁡ ( H ) − = Θ S , T # Fitt_(R)(Sel_(S)^(T)(H)^(-))=(Theta_(S,T)^(#))\operatorname{Fitt}_{R}\left(\operatorname{Sel}_{S}^{T}(H)^{-}\right)=\left(\Theta_{S, T}^{\#}\right)FittR⁡(SelST⁡(H)−)=(ΘS,T#)
We have proven a version of this result with altered sets S S SSS and T T TTT. Fix an odd prime p p ppp and put R p = Z p [ G ] R p = Z p [ G ] − R_(p)=Z_(p)[G]^(-)R_{p}=\mathbf{Z}_{p}[G]^{-}Rp=Zp[G]−. Define
Σ = S { v S : v p } Σ = S ∖ { v ∈ S : v ∤ p } Sigma=S\\{v in S:v∤p}\Sigma=S \backslash\{v \in S: v \nmid p\}Σ=S∖{v∈S:v∤p}
and
Σ = T { v S : v p } Σ ′ = T ∪ { v ∈ S : v ∤ p } Sigma^(')=T uu{v in S:v∤p}\Sigma^{\prime}=T \cup\{v \in S: v \nmid p\}Σ′=T∪{v∈S:v∤p}
Theorem 5.6 ([17, THEOREM 3.3]). Let Sel Σ Σ ( H ) p = Sel Σ Σ ( H ) Z [ G ] R p Sel Σ Σ ′ ⁡ ( H ) p − = Sel Σ Σ ′ ⁡ ( H ) ⊗ Z [ G ] R p Sel_(Sigma)^(Sigma^('))(H)_(p)^(-)=Sel_(Sigma)^(Sigma^('))(H)ox_(Z[G])R_(p)\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}=\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H) \otimes_{\mathbf{Z}[G]} R_{p}SelΣΣ′⁡(H)p−=SelΣΣ′⁡(H)⊗Z[G]Rp. We have
Fitt R p ( Sel Σ Σ ( H ) p ) = ( Θ Σ , Σ # ) Fitt R p ⁡ Sel Σ Σ ′ ⁡ ( H ) p − = Θ Σ , Σ ′ # Fitt_(R_(p))(Sel_(Sigma)^(Sigma^('))(H)_(p)^(-))=(Theta_(Sigma,Sigma^('))^(#))\operatorname{Fitt}_{R_{p}}\left(\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}\right)=\left(\Theta_{\Sigma, \Sigma^{\prime}}^{\#}\right)FittRp⁡(SelΣΣ′⁡(H)p−)=(ΘΣ,Σ′#)
It turns out that Theorem 5.6 is strong enough to imply Kurihara's conjecture (Theorem 5.1). The key point is that there is a short exact sequence
(5.1) 0 Sel Σ T ( H ) Sel Σ Σ ( H ) w S H ( ( O H / w ) ) , 0 , (5.1) 0 ⟶ Sel Σ T ⁡ ( H ) − ⟶ Sel Σ Σ ′ ⁡ ( H ) − ⟶ ∏ w ∈ S H ′   O H / w ∗ ∨ , − ⟶ 0 , {:(5.1)0longrightarrowSel_(Sigma)^(T)(H)^(-)longrightarrowSel_(Sigma)^(Sigma^('))(H)^(-)longrightarrowprod_(w inS_(H)^('))((O_(H)//w)^(**))^(vv,-)longrightarrow0",":}\begin{equation*} 0 \longrightarrow \operatorname{Sel}_{\Sigma}^{T}(H)^{-} \longrightarrow \operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)^{-} \longrightarrow \prod_{w \in S_{H}^{\prime}}\left(\left(O_{H} / w\right)^{*}\right)^{\vee,-} \longrightarrow 0, \tag{5.1} \end{equation*}(5.1)0⟶SelΣT⁡(H)−⟶SelΣΣ′⁡(H)−⟶∏w∈SH′((OH/w)∗)∨,−⟶0,
from which one deduces (see [ 17 [ 17 [17[17[17, THEOREM 3.7 ] ] ]]] )
Fitt R p ( Sel Σ T ( H ) p ) = ( Θ Σ , T # ) v S r a m , v p ( N I v , 1 σ v e v ) . Fitt R p ⁡ Sel Σ T ⁡ ( H ) p − = Θ Σ , T # ∏ v ∈ S r a m , v ∤ p   N I v , 1 − σ v e v . Fitt_(R_(p))(Sel_(Sigma)^(T)(H)_(p)^(-))=(Theta_(Sigma,T)^(#))prod_(v inS_(ram),v∤p)((N)I_(v),1-sigma_(v)e_(v)).\operatorname{Fitt}_{R_{p}}\left(\operatorname{Sel}_{\Sigma}^{T}(H)_{p}^{-}\right)=\left(\Theta_{\Sigma, T}^{\#}\right) \prod_{v \in S_{\mathrm{ram}}, v \nmid p}\left(\mathrm{~N} I_{v}, 1-\sigma_{v} e_{v}\right) .FittRp⁡(SelΣT⁡(H)p−)=(ΘΣ,T#)∏v∈Sram,v∤p( NIv,1−σvev).
Since Sel S T ( H ) C l T ( H ) , v Sel S ∞ T ⁡ ( H ) − ≅ C l T ( H ) − , v Sel_(S_(oo))^(T)(H)^(-)~=Cl^(T)(H)^(-,v)\operatorname{Sel}_{S_{\infty}}^{T}(H)^{-} \cong \mathrm{Cl}^{T}(H)^{-, v}SelS∞T⁡(H)−≅ClT(H)−,v, it then remains to calculate the effect of removing the primes v S ram , v p v ∈ S ram  , v ∣ p v inS_("ram "),v∣pv \in S_{\text {ram }}, v \mid pv∈Sram ,v∣p from Σ Î£ Sigma\SigmaΣ. This is a delicate process using functorial properties of the Ritter-Weiss modules discussed in § 6 § 6 §6\S 6§6, and one obtains (see [17, APPENDIX B]) the desired result
Fitt R p ( Sel S T ( H ) p ) = ( Θ S , T # ) v S r a m ( N I v , 1 σ v e v ) Fitt R p ⁡ Sel S ∞ T ⁡ ( H ) p − = Θ S ∞ , T # ∏ v ∈ S r a m   N I v , 1 − σ v e v Fitt_(R_(p))(Sel_(S_(oo))^(T)(H)_(p)^(-))=(Theta_(S_(oo),T)^(#))prod_(v inS_(ram))(NI_(v),1-sigma_(v)e_(v))\operatorname{Fitt}_{R_{p}}\left(\operatorname{Sel}_{S_{\infty}}^{T}(H)_{p}^{-}\right)=\left(\Theta_{S_{\infty}, T}^{\#}\right) \prod_{v \in S_{\mathrm{ram}}}\left(\mathrm{N} I_{v}, 1-\sigma_{v} e_{v}\right)FittRp⁡(SelS∞T⁡(H)p−)=(ΘS∞,T#)∏v∈Sram(NIv,1−σvev)

6. RITTER-WEISS MODULES AND RIBET'S METHOD

In this section we summarize the proof of Theorem 5.6.

6.1. Ritter-Weiss modules

The Z [ G ] Z [ G ] Z[G]\mathbf{Z}[G]Z[G]-module that shows up in our constructions with modular forms is a certain transpose of Sel S T ( H ) Sel S T ⁡ ( H ) Sel_(S)^(T)(H)\operatorname{Sel}_{S}^{T}(H)SelST⁡(H) in the sense of Jannsen [31], denoted S T ( H ) ∇ S T ( H ) grad_(S)^(T)(H)\nabla_{S}^{T}(H)∇ST(H). This module was originally defined by Ritter and Weiss in the foundational paper [40] without the smoothing set T T TTT. We incorporated the smoothing set T T TTT and established some additional properties of S T ( H ) ∇ S T ( H ) grad_(S)^(T)(H)\nabla_{S}^{T}(H)∇ST(H) in [17, APPENDIX A]. To describe these properties, we work over R p = Z p [ G ] R p = Z p [ G ] − R_(p)=Z_(p)[G]^(-)R_{p}=\mathbf{Z}_{p}[G]^{-}Rp=Zp[G]−and consider finite disjoint sets Σ , Σ Î£ , Σ ′ Sigma,Sigma^(')\Sigma, \Sigma^{\prime}Σ,Σ′ satisfying the following:
  • Σ S Σ ⊃ S ∞ Sigma supS_(oo)\Sigma \supset S_{\infty}Σ⊃S∞ and Σ Σ S ram Σ ∪ Σ ′ ⊃ S ram  Sigma uuSigma^(')supS_("ram ")\Sigma \cup \Sigma^{\prime} \supset S_{\text {ram }}Σ∪Σ′⊃Sram .
  • Σ Î£ ′ Sigma^(')\Sigma^{\prime}Σ′ satisfies the condition (2) on T T TTT in Section 2.
  • The primes in Σ S ram Σ ′ ∩ S ram  Sigma^(')nnS_("ram ")\Sigma^{\prime} \cap S_{\text {ram }}Σ′∩Sram  have residue characteristic p â„“ ≠ p â„“!=p\ell \neq pℓ≠p.
Note that the pair ( S , T ) ( S , T ) (S,T)(S, T)(S,T) from Section 2 and the pair ( Σ , Σ ) Σ , Σ ′ (Sigma,Sigma^('))\left(\Sigma, \Sigma^{\prime}\right)(Σ,Σ′) considered in Section 5.3 both satisfy these conditions. The module Σ Σ ( H ) p = Σ Σ ( H ) Z [ G ] R p ∇ Σ Σ ′ ( H ) p − = ∇ Σ Σ ′ ( H ) ⊗ Z [ G ] R p grad_(Sigma)^(Sigma^('))(H)_(p)^(-)=grad_(Sigma)^(Sigma^('))(H)ox_(Z[G])R_(p)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}=\nabla_{\Sigma}^{\Sigma^{\prime}}(H) \otimes_{\mathbf{Z}[G]} R_{p}∇ΣΣ′(H)p−=∇ΣΣ′(H)⊗Z[G]Rp satisfies the following:
  • There is a short exact sequence of R p R p R_(p)R_{p}Rp-modules
(6.1) 0 C l Σ Σ ( H ) p Σ Σ ( H ) p ( X H Σ ) p 0 (6.1) 0 ⟶ C l Σ Σ ′ ( H ) p − ⟶ ∇ Σ Σ ′ ( H ) p − ⟶ X H Σ p − ⟶ 0 {:(6.1)0longrightarrowCl_(Sigma)^(Sigma^('))(H)_(p)^(-)longrightarrowgrad_(Sigma)^(Sigma^('))(H)_(p)^(-)longrightarrow(X_(H_(Sigma)))_(p)^(-)longrightarrow0:}\begin{equation*} 0 \longrightarrow \mathrm{Cl}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-} \longrightarrow \nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-} \longrightarrow\left(X_{H_{\Sigma}}\right)_{p}^{-} \longrightarrow 0 \tag{6.1} \end{equation*}(6.1)0⟶ClΣΣ′(H)p−⟶∇ΣΣ′(H)p−⟶(XHΣ)p−⟶0
Here C l Σ Σ ( H ) C l Σ Σ ′ ( H ) Cl_(Sigma)^(Sigma^('))(H)\mathrm{Cl}_{\Sigma}^{\Sigma^{\prime}}(H)ClΣΣ′(H) denotes the quotient of C l Σ ( H ) C l Σ ′ ( H ) Cl^(Sigma^('))(H)\mathrm{Cl}^{\Sigma^{\prime}}(H)ClΣ′(H) by the image of the primes in Σ H Σ H Sigma_(H)\Sigma_{H}ΣH.
  • Given a R p R p R_(p)R_{p}Rp-module B B BBB, a surjective R p R p R_(p)R_{p}Rp-module homomorphism
(6.2) Σ Σ ( H ) p B (6.2) ∇ Σ Σ ′ ( H ) p − → B {:(6.2)grad_(Sigma)^(Sigma^('))(H)_(p)^(-)rarr B:}\begin{equation*} \nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-} \rightarrow B \tag{6.2} \end{equation*}(6.2)∇ΣΣ′(H)p−→B
is equivalent to the data of a cocycle κ Z 1 ( G F , B ) κ ∈ Z 1 G F , B kappa inZ^(1)(G_(F),B)\kappa \in Z^{1}\left(G_{F}, B\right)κ∈Z1(GF,B) and a collection of elements x v B x v ∈ B x_(v)in Bx_{v} \in Bxv∈B for v Σ v ∈ Σ v in Sigmav \in \Sigmav∈Σ satsifying the following conditions:
  • The cohomology class [ κ ] H 1 ( G F , B ) [ κ ] ∈ H 1 G F , B [kappa]inH^(1)(G_(F),B)[\kappa] \in H^{1}\left(G_{F}, B\right)[κ]∈H1(GF,B) is unramified outside Σ Î£ ′ Sigma^(')\Sigma^{\prime}Σ′, tamely ramified at Σ Î£ ′ Sigma^(')\Sigma^{\prime}Σ′, and locally trivial at Σ Î£ Sigma\SigmaΣ.
  • The x v x v x_(v)x_{v}xv provide local trivializations at Σ : κ ( σ ) = ( σ 1 ) x v Σ : κ ( σ ) = ( σ − 1 ) x v Sigma:kappa(sigma)=(sigma-1)x_(v)\Sigma: \kappa(\sigma)=(\sigma-1) x_{v}Σ:κ(σ)=(σ−1)xv for σ G v σ ∈ G v sigma inG_(v)\sigma \in G_{v}σ∈Gv.
  • The x v x v x_(v)x_{v}xv along with the image of κ κ kappa\kappaκ generate the module B B BBB over R p R p R_(p)R_{p}Rp.
The tuples ( κ , { x v } ) κ , x v (kappa,{x_(v)})\left(\kappa,\left\{x_{v}\right\}\right)(κ,{xv}) are taken modulo the natural notion of coboundary, i.e., ( κ , { x v } ) ( κ + d x , { x v + x } ) κ , x v ∼ κ + d x , x v + x (kappa,{x_(v)})∼(kappa+dx,{x_(v)+x})\left(\kappa,\left\{x_{v}\right\}\right) \sim\left(\kappa+d x,\left\{x_{v}+x\right\}\right)(κ,{xv})∼(κ+dx,{xv+x}) for x B x ∈ B x in Bx \in Bx∈B.
  • The module Σ Σ ( H ) p ∇ Σ Σ ′ ( H ) p − grad_(Sigma)^(Sigma^('))(H)_(p)^(-)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}∇ΣΣ′(H)p−has a quadratic presentation, i.e., there exists an exact sequence of R p R p R_(p)R_{p}Rp-modules
(6.3) M 1 A M 2 Σ Σ ( H ) p 0 (6.3) M 1 → A M 2 ⟶ ∇ Σ Σ ′ ( H ) p − ⟶ 0 {:(6.3)M_(1)rarr"A"M_(2)longrightarrowgrad_(Sigma)^(Sigma^('))(H)_(p)^(-)longrightarrow0:}\begin{equation*} M_{1} \xrightarrow{A} M_{2} \longrightarrow \nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-} \longrightarrow 0 \tag{6.3} \end{equation*}(6.3)M1→AM2⟶∇ΣΣ′(H)p−⟶0
where M 1 M 1 M_(1)M_{1}M1 and M 2 M 2 M_(2)M_{2}M2 are free of the same finite rank.
  • The module Σ Σ ( H ) p ∇ Σ Σ ′ ( H ) p − grad_(Sigma)^(Sigma^('))(H)_(p)^(-)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}∇ΣΣ′(H)p−is a transpose of Sel Σ Σ ( H ) p Sel Σ Σ ′ ⁡ ( H ) p − Sel_(Sigma)^(Sigma^('))(H)_(p)^(-)\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}SelΣΣ′⁡(H)p−, i.e., for a suitable quadratic presentation (6.3) of Σ Σ ( H ) p ∇ Σ Σ ′ ( H ) p − grad_(Sigma)^(Sigma^('))(H)_(p)^(-)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}∇ΣΣ′(H)p−, the cokernel of the induced map
(6.4) Hom R p ( M 2 , R p ) A T , # Hom R p ( M 1 , R p ) (6.4) Hom R p ⁡ M 2 , R p → A T , # Hom R p ⁡ M 1 , R p {:(6.4)Hom_(R_(p))(M_(2),R_(p))rarr"A^(T,#)"Hom_(R_(p))(M_(1),R_(p)):}\begin{equation*} \operatorname{Hom}_{R_{p}}\left(M_{2}, R_{p}\right) \xrightarrow{A^{T, \#}} \operatorname{Hom}_{R_{p}}\left(M_{1}, R_{p}\right) \tag{6.4} \end{equation*}(6.4)HomRp⁡(M2,Rp)→AT,#HomRp⁡(M1,Rp)
is isomorphic to Sel Σ Σ ( H ) p Sel Σ Σ ′ ⁡ ( H ) p − Sel_(Sigma)^(Sigma^('))(H)_(p)^(-)\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}SelΣΣ′⁡(H)p−. Here we follow our convention of giving Hom spaces the contragradient G G GGG-action.
The quadratic presentation property (6.3) implies that Fitt R p ( Σ Σ ( H ) p ) = det ( A ) Fitt R p ⁡ ∇ Σ Σ ′ ( H ) p − = det ⁡ ( A ) Fitt_(R_(p))(grad_(Sigma)^(Sigma^('))(H)_(p)^(-))=det(A)\operatorname{Fitt}_{R_{p}}\left(\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}\right)=\operatorname{det}(A)FittRp⁡(∇ΣΣ′(H)p−)=det⁡(A) is principal. The transpose property (6.4) implies that
(6.5) Fitt R p ( Σ Σ ( H ) p ) = Fitt R p ( Sel Σ Σ ( H ) p ) # (6.5) Fitt R p ⁡ ∇ Σ Σ ′ ( H ) p − = Fitt R p ⁡ Sel Σ Σ ′ ⁡ ( H ) p − # {:(6.5)Fitt_(R_(p))(grad_(Sigma)^(Sigma^('))(H)_(p)^(-))=Fitt_(R_(p))(Sel_(Sigma)^(Sigma^('))(H)_(p)^(-))^(#):}\begin{equation*} \operatorname{Fitt}_{R_{p}}\left(\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}\right)=\operatorname{Fitt}_{R_{p}}\left(\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}\right)^{\#} \tag{6.5} \end{equation*}(6.5)FittRp⁡(∇ΣΣ′(H)p−)=FittRp⁡(SelΣΣ′⁡(H)p−)#
Theorem 5.6 is therefore equivalent to
(6.6) Fitt R p ( Σ Σ ( H ) p ) = ( Θ Σ , Σ ) (6.6) Fitt R p ⁡ ∇ Σ Σ ′ ( H ) p − = Θ Σ , Σ ′ {:(6.6)Fitt_(R_(p))(grad_(Sigma)^(Sigma^('))(H)_(p)^(-))=(Theta_(Sigma,Sigma^('))):}\begin{equation*} \operatorname{Fitt}_{R_{p}}\left(\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}\right)=\left(\Theta_{\Sigma, \Sigma^{\prime}}\right) \tag{6.6} \end{equation*}(6.6)FittRp⁡(∇ΣΣ′(H)p−)=(ΘΣ,Σ′)
We now fix ( Σ , Σ ) Σ , Σ ′ (Sigma,Sigma^('))\left(\Sigma, \Sigma^{\prime}\right)(Σ,Σ′) to be the pair defined in Section 5.3. In the remainder of this section we outline how (6.6) is proved using Ribet's method. Throughout, an unadorned Θ Î˜ Theta\ThetaΘ denotes Θ Σ , Σ ( Θ Σ , Σ ′ Theta_(Sigma,Sigma^('))(:}\Theta_{\Sigma, \Sigma^{\prime}}\left(\right.ΘΣ,Σ′( and Θ # Θ # Theta^(#)\Theta^{\#}Θ# denotes Θ Σ , Σ # Θ Σ , Σ ′ # Theta_(Sigma,Sigma^('))^(#)\Theta_{\Sigma, \Sigma^{\prime}}^{\#}ΘΣ,Σ′#.

6.2. Inclusion implies equality

An interesting feature of Ribet's method is that it tends to prove an inclusion in one direction, that of an algebraically defined ideal contained within an analytically defined ideal. In our setting, we use it to prove
(6.7) Fitt R p ( Σ Σ ( H ) p ) ( Θ ) , equivalently, Fitt R p ( Sel Σ Σ ( H ) p ) ( Θ # ) (6.7) Fitt R p ⁡ ∇ Σ Σ ′ ( H ) p − ⊂ ( Θ ) ,  equivalently,  Fitt R p ⁡ Sel Σ Σ ′ ⁡ ( H ) p − ⊂ Θ # {:(6.7)Fitt_(R_(p))(grad_(Sigma)^(Sigma^('))(H)_(p)^(-))sub(Theta)","quad" equivalently, "quadFitt_(R_(p))(Sel_(Sigma)^(Sigma^('))(H)_(p)^(-))sub(Theta^(#)):}\begin{equation*} \operatorname{Fitt}_{R_{p}}\left(\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}\right) \subset(\Theta), \quad \text { equivalently, } \quad \operatorname{Fitt}_{R_{p}}\left(\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}\right) \subset\left(\Theta^{\#}\right) \tag{6.7} \end{equation*}(6.7)FittRp⁡(∇ΣΣ′(H)p−)⊂(Θ), equivalently, FittRp⁡(SelΣΣ′⁡(H)p−)⊂(Θ#)
We then employ an analytic argument to show that this inclusion is an equality. It is important to note that the inclusion (6.7) is the reverse direction of that required by the Brumer-Stark and Strong Brumer-Stark conjectures. It is therefore essential for our approach that one actually has the statement of an equality rather than just an inclusion (and an analytic argument to deduce the equality from the reverse inclusion). For this reason, the conjecture of Burns stated in Section 5.3 (more precisely the analog of it stated in Theorem 5.6) plays an essential role in our strategy.
Let us describe the analytic argument in the special case Σ = S Σ = S ∞ Sigma=S_(oo)\Sigma=S_{\infty}Σ=S∞, i.e. there are no primes above p p ppp ramified in H / F H / F H//FH / FH/F. In this case
(6.8) Sel Σ Σ ( H ) p C l Σ ( H ) p (6.8) Sel Σ Σ ′ ⁡ ( H ) p − ≅ C l Σ ′ ( H ) p − {:(6.8)Sel_(Sigma)^(Sigma^('))(H)_(p)^(-)~=Cl^(Sigma^('))(H)_(p)^(-):}\begin{equation*} \operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-} \cong \mathrm{Cl}^{\Sigma^{\prime}}(H)_{p}^{-} \tag{6.8} \end{equation*}(6.8)SelΣΣ′⁡(H)p−≅ClΣ′(H)p−
is finite and Θ Î˜ Theta\ThetaΘ is a non-zero-divisor. Using (6.7), write
(6.9) Fitt R p ( Sel Σ Σ ( H ) p ) = ( Θ # z ) for some z R p (6.9) Fitt R p ⁡ Sel Σ Σ ′ ⁡ ( H ) p − = Θ # â‹… z  for some  z ∈ R p {:(6.9)Fitt_(R_(p))(Sel_(Sigma)^(Sigma^('))(H)_(p)^(-))=(Theta^(#)*z)quad" for some "z inR_(p):}\begin{equation*} \operatorname{Fitt}_{R_{p}}\left(\operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}\right)=\left(\Theta^{\#} \cdot z\right) \quad \text { for some } z \in R_{p} \tag{6.9} \end{equation*}(6.9)FittRp⁡(SelΣΣ′⁡(H)p−)=(Θ#â‹…z) for some z∈Rp
We must show that z R p z ∈ R p ∗ z inR_(p)^(**)z \in R_{p}^{*}z∈Rp∗. An elementary argument (see [17, $2.3]) shows that (6.9) implies
(6.10) # Sel Σ Σ ( H ) p = ( χ G ^ χ odd χ ( Θ # z ) ) p (6.10) # Sel Σ Σ ′ ⁡ ( H ) p − = ∏ χ ∈ G ^ χ  odd    χ Θ # â‹… z p {:(6.10)#Sel_(Sigma)^(Sigma^('))(H)_(p)^(-)=(prod_({:[chi in hat(G)],[chi" odd "]:})chi(Theta^(#)*z))_(p):}\begin{equation*} \# \operatorname{Sel}_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}=\left(\prod_{\substack{\chi \in \hat{G} \\ \chi \text { odd }}} \chi\left(\Theta^{\#} \cdot z\right)\right)_{p} \tag{6.10} \end{equation*}(6.10)#SelΣΣ′⁡(H)p−=(∏χ∈G^χ odd χ(Θ#â‹…z))p
where the subscript p p ppp on the right denotes the p p ppp-power part of an integer. Yet the analytic class number formula implies (see [ 17 , $ 2.1 ] [ 17 , $ 2.1 ] [17,$2.1][17, \$ 2.1][17,$2.1] )
(6.11) χ G ^ χ odd χ ( Θ # ) = χ G ^ χ odd L Σ , Σ ( χ , 0 ) # C l Σ ( H ) (6.11) ∏ χ ∈ G ^ χ  odd    χ Θ # = ∏ χ ∈ G ^ χ  odd    L Σ , Σ ′ ( χ , 0 ) ≐ # C l Σ ′ ( H ) − {:(6.11)prod_({:[chi in hat(G)],[chi" odd "]:})chi(Theta^(#))=prod_({:[chi in hat(G)],[chi" odd "]:})L_(Sigma,Sigma^('))(chi","0)≐#Cl^(Sigma^('))(H)^(-):}\begin{equation*} \prod_{\substack{\chi \in \hat{G} \\ \chi \text { odd }}} \chi\left(\Theta^{\#}\right)=\prod_{\substack{\chi \in \hat{G} \\ \chi \text { odd }}} L_{\Sigma, \Sigma^{\prime}}(\chi, 0) \doteq \# \mathrm{Cl}^{\Sigma^{\prime}}(H)^{-} \tag{6.11} \end{equation*}(6.11)∏χ∈G^χ odd χ(Θ#)=∏χ∈G^χ odd LΣ,Σ′(χ,0)≐#ClΣ′(H)−
where ≐ ≐\doteq≐ denotes equality up to a power of 2 . Combining (6.8), (6.10), and (6.11), one finds that χ ( z ) χ ( z ) chi(z)\chi(z)χ(z) is a p p ppp-adic unit for each odd character χ χ chi\chiχ. It follows that z R p z ∈ R p ∗ z inR_(p)^(**)z \in R_{p}^{*}z∈Rp∗ as desired.
The generalization of this argument to arbitrary Σ Î£ Sigma\SigmaΣ requires a delicate induction and is described in [ 17 , $ 5 ] [ 17 , $ 5 ] [17,$5][17, \$ 5][17,$5].

6.3. Ribet's method

We now describe our implementation of Ribet's method to prove the inclusion (6.7). The idea is to use the Galois representations associated to Hilbert modular forms to construct an R p R p R_(p)R_{p}Rp-module M M MMM and a surjection Σ Σ ( H ) p M ∇ Σ Σ ′ ( H ) p − → M grad_(Sigma)^(Sigma^('))(H)_(p)^(-)rarr M\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-} \rightarrow M∇ΣΣ′(H)p−→M such that Fitt R p ( M ) ( Θ ) Fitt R p ⁡ ( M ) ⊂ ( Θ ) Fitt_(R_(p))(M)sub(Theta)\operatorname{Fitt}_{R_{p}}(M) \subset(\Theta)FittRp⁡(M)⊂(Θ). The properties of Fitting ideals imply that (6.7) follows from the existence of such a surjection. As described in (6.2), a surjection from Σ Σ ( H ) p ∇ Σ Σ ′ ( H ) p − grad_(Sigma)^(Sigma^('))(H)_(p)^(-)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}∇ΣΣ′(H)p−can be constructed by defining a cohomology class [ κ ] H 1 ( G F , M ) [ κ ] ∈ H 1 G F , M [kappa]inH^(1)(G_(F),M)[\kappa] \in H^{1}\left(G_{F}, M\right)[κ]∈H1(GF,M) satisfying certain local conditions along with local trivializations at the places in Σ Î£ Sigma\SigmaΣ.
Ribet's method was described in great detail by Mazur in a beautiful article written for the celebration of Ribet's 60th birthday [36]. We borrow from this the following schematic diagram demonstrating the general path one follows to link L L LLL-values (in our case, the Stickelberger element Θ Î˜ Theta\ThetaΘ ) to class groups (in our case, the Ritter-Weiss module Σ Σ ( H ) p ∇ Σ Σ ′ ( H ) p − grad_(Sigma)^(Sigma^('))(H)_(p)^(-)\nabla_{\Sigma}^{\Sigma^{\prime}}(H)_{p}^{-}∇ΣΣ′(H)p−).
Let us now trace this path in our application.

6.3.1. L L LLL-functions to Eisenstein series

The key connection between L L LLL-functions and modular forms in Ribet's method is that L L LLL-functions appear as constant terms of Eisenstein series. We now define the space of modular forms in which our Stickelberger element Θ Î˜ Theta\ThetaΘ appears.
Let k > 1 k > 1 k > 1k>1k>1 be an integer such that
k 1 ( mod ( p 1 ) p N ) k ≡ 1 mod ( p − 1 ) p N k-=1(mod(p-1)p^(N))k \equiv 1\left(\bmod (p-1) p^{N}\right)k≡1(mod(p−1)pN)
for a large value of N N NNN. Let n O F n ⊂ O F nsubO_(F)\mathfrak{n} \subset O_{F}n⊂OF denote the conductor of H / F H / F H//FH / FH/F. Let M k ( n ; Z ) M k ( n ; Z ) M_(k)(n;Z)M_{k}(\mathfrak{n} ; \mathbf{Z})Mk(n;Z) denote the group of Hilbert modular forms for F F FFF of level n n n\mathfrak{n}n with Fourier coefficients in Z Z Z\mathbf{Z}Z. For each odd character χ χ chi\chiχ of G G GGG valued in C p C p ∗ C_(p)^(**)\mathbf{C}_{p}^{*}Cp∗, let
M k ( n , χ ) M k ( n ; Z ) C p M k ( n , χ ) ⊂ M k ( n ; Z ) ⊗ C p M_(k)(n,chi)subM_(k)(n;Z)oxC_(p)M_{k}(\mathfrak{n}, \chi) \subset M_{k}(\mathfrak{n} ; \mathbf{Z}) \otimes \mathbf{C}_{p}Mk(n,χ)⊂Mk(n;Z)⊗Cp
denote the subspace of forms of nebentypus χ χ chi\chiχ. Let
χ : G F G R p χ : G F → G → R p ∗ chi:G_(F)rarr G rarrR_(p)^(**)\chi: G_{F} \rightarrow G \rightarrow R_{p}^{*}χ:GF→G→Rp∗
denote the canonical character, where the first arrow is projection and the second is induced by G Z [ G ] G ↪ Z [ G ] ∗ G↪Z[G]^(**)G \hookrightarrow \mathbf{Z}[G]^{*}G↪Z[G]∗.
Definition 6.1. The space M k ( n , χ ; R p ) M k n , χ ; R p M_(k)(n,chi;R_(p))M_{k}\left(\mathfrak{n}, \chi ; R_{p}\right)Mk(n,χ;Rp) of group-ring valued Hilbert modular forms of weight k k kkk and level n n n\mathfrak{n}n over R p R p R_(p)R_{p}Rp consists of those f M k ( n ; Z ) R p f ∈ M k ( n ; Z ) ⊗ R p f inM_(k)(n;Z)oxR_(p)f \in M_{k}(\mathfrak{n} ; \mathbf{Z}) \otimes R_{p}f∈Mk(n;Z)⊗Rp such that χ ( f ) M k ( n , χ ) χ ( f ) ∈ M k ( n , χ ) chi(f)inM_(k)(n,chi)\chi(f) \in M_{k}(\mathfrak{n}, \chi)χ(f)∈Mk(n,χ) for each odd character χ χ chi\chiχ. Let S k ( n , χ ; R p ) S k n , χ ; R p S_(k)(n,chi;R_(p))S_{k}\left(\mathfrak{n}, \chi ; R_{p}\right)Sk(n,χ;Rp) denote the subspace of cusp forms. We define M k ( n , χ ; Frac ( R p ) ) M k n , χ ; Frac ⁡ R p M_(k)(n,chi;Frac(R_(p)))M_{k}\left(\mathfrak{n}, \chi ; \operatorname{Frac}\left(R_{p}\right)\right)Mk(n,χ;Frac⁡(Rp)) and S k ( n , χ ; Frac ( R p ) ) S k n , χ ; Frac ⁡ R p S_(k)(n,chi;Frac(R_(p)))S_{k}\left(\mathfrak{n}, \chi ; \operatorname{Frac}\left(R_{p}\right)\right)Sk(n,χ;Frac⁡(Rp)) similarly.
Hilbert modular forms f f fff are determined by their Fourier coefficients c ( m , f ) c ( m , f ) c(m,f)c(\mathfrak{m}, f)c(m,f) indexed by the nonzero ideals m O F m ⊂ O F msubO_(F)\mathfrak{m} \subset O_{F}m⊂OF and their constant terms c λ ( 0 , f ) c λ ( 0 , f ) c_(lambda)(0,f)c_{\lambda}(0, f)cλ(0,f) indexed by λ C l + ( F ) λ ∈ C l + ( F ) lambda inCl^(+)(F)\lambda \in \mathrm{Cl}^{+}(F)λ∈Cl+(F), the narrow class group of F F FFF. For odd k 1 k ≥ 1 k >= 1k \geq 1k≥1, there is an Eisenstein series E k ( χ , 1 ) M k ( n , χ ) E k ( χ , 1 ) ∈ M k ( n , χ ) E_(k)(chi,1)inM_(k)(n,chi)E_{k}(\chi, 1) \in M_{k}(\mathfrak{n}, \chi)Ek(χ,1)∈Mk(n,χ) whose Fourier coefficients are given by
c ( m , E k ( χ , 1 ) ) = a m ( m / a , n ) = 1 χ ( m / a ) N a k 1 c m , E k ( χ , 1 ) = ∑ a ∣ m ( m / a , n ) = 1   χ ( m / a ) N a k − 1 c(m,E_(k)(chi,1))=sum_({:[a∣m],[(m//a","n)=1]:})chi(m//a)Na^(k-1)c\left(\mathfrak{m}, E_{k}(\chi, 1)\right)=\sum_{\substack{a \mid \mathfrak{m} \\(\mathfrak{m} / \mathfrak{a}, \mathfrak{n})=1}} \chi(\mathfrak{m} / \mathfrak{a}) \mathrm{Na}^{k-1}